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<h1 class="ltx_title ltx_title_document">
<span id="id12.id1" class="ltx_ERROR undefined">\logo</span> ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving</h1>
<div class="ltx_authors">
<span class="ltx_creator ltx_role_author">
<span class="ltx_personname">
Zhibin Gou<sup id="id13.12.id1" class="ltx_sup"><span id="id13.12.id1.1" class="ltx_text ltx_font_italic">1,2</span></sup> ,
Zhihong Shao<sup id="id14.13.id2" class="ltx_sup"><span id="id14.13.id2.1" class="ltx_text ltx_font_italic">1,2∗</span></sup>,
Yeyun Gong<sup id="id15.14.id3" class="ltx_sup"><span id="id15.14.id3.1" class="ltx_text ltx_font_italic">2†</span></sup>,
Yelong Shen<sup id="id16.15.id4" class="ltx_sup"><span id="id16.15.id4.1" class="ltx_text ltx_font_italic">3</span></sup>
<br class="ltx_break"><span id="id8.8.4" class="ltx_text ltx_font_bold">
Yujiu Yang<sup id="id8.8.4.1" class="ltx_sup"><span id="id8.8.4.1.1" class="ltx_text ltx_font_medium ltx_font_italic">1</span></sup> ,
Minlie Huang<sup id="id8.8.4.2" class="ltx_sup"><span id="id8.8.4.2.1" class="ltx_text ltx_font_medium ltx_font_italic">1†</span></sup>,
Nan Duan<sup id="id8.8.4.3" class="ltx_sup"><span id="id8.8.4.3.1" class="ltx_text ltx_font_medium ltx_font_italic">2</span></sup>,
Weizhu Chen<sup id="id8.8.4.4" class="ltx_sup"><span id="id8.8.4.4.1" class="ltx_text ltx_font_medium ltx_font_italic">3</span></sup></span>

<br class="ltx_break"><sup id="id17.16.id5" class="ltx_sup">1</sup>Tsinghua University
<sup id="id18.17.id6" class="ltx_sup">2</sup>Microsoft Research
<sup id="id19.18.id7" class="ltx_sup">3</sup>Microsoft Azure AI

<br class="ltx_break"><span id="id20.19.id8" class="ltx_text ltx_font_typewriter">{gzb22,szh19}@mails.tsinghua.edu.cn
<br class="ltx_break">{yegong,yeshe,nanduan,wzchen}@microsoft.com</span>
</span><span class="ltx_author_notes">Equal contribution. See Contributions section for details. Work done during an internship at Microsoft.<span id="id21.20.id1" class="ltx_text ltx_font_bold">Corresponding authors.</span></span></span>
</div>

<div class="ltx_abstract">
<h6 class="ltx_title ltx_title_abstract">Abstract</h6>
<p id="id22.id1" class="ltx_p">Large language models have made significant progress in various language tasks, yet they still struggle with complex mathematics.
In this paper, we propose <span id="id22.id1.1" class="ltx_text ltx_font_smallcaps">ToRA</span>, a series of <span id="id22.id1.2" class="ltx_text ltx_framed ltx_framed_underline">To</span>ol-integrated <span id="id22.id1.3" class="ltx_text ltx_framed ltx_framed_underline">R</span>easoning <span id="id22.id1.4" class="ltx_text ltx_framed ltx_framed_underline">A</span>gents designed to solve challenging mathematical problems by seamlessly integrating natural language reasoning with the utilization of external tools (e.g., computation libraries and symbolic solvers), thereby amalgamating the analytical prowess of language and the computational efficiency of tools.
To train <span id="id22.id1.5" class="ltx_text ltx_font_smallcaps">ToRA</span>, we curate interactive tool-use trajectories on mathematical datasets, apply imitation learning on the annotations, and propose output space shaping to further refine models’ reasoning behavior.
As a result, <span id="id22.id1.6" class="ltx_text ltx_font_smallcaps">ToRA</span> models significantly outperform open-source models on 10 mathematical reasoning datasets across all scales with 13%-19% absolute improvements on average.
Notably, <span id="id22.id1.7" class="ltx_text ltx_font_smallcaps">ToRA</span>-7B reaches 44.6% on the competition-level dataset MATH, surpassing the best open-source model WizardMath-70B by 22% absolute.
<span id="id22.id1.8" class="ltx_text ltx_font_smallcaps">ToRA-Code</span>-34B is also the first open-source model that achieves an accuracy exceeding 50% on MATH, which significantly outperforms GPT-4’s CoT result, and is competitive with GPT-4 solving problems with programs.
Additionally, we conduct a comprehensive analysis of the benefits and remaining challenges of tool interaction for mathematical reasoning, providing valuable insights for future research<span id="footnote1" class="ltx_note ltx_role_footnote"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>Code and models are available at <a target="_blank" href="https://github.com/microsoft/ToRA" title="" class="ltx_ref ltx_url ltx_font_typewriter">https://github.com/microsoft/ToRA</a>.</span></span></span>.</p>
</div>
<figure id="S0.F1" class="ltx_figure"><img src="/html/2309.17452/assets/x1.png" id="S0.F1.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="414" height="152" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>Comparing <span id="S0.F1.4.1" class="ltx_text ltx_font_smallcaps">ToRA</span> with baselines on LLaMA-2 base models from 7B to 70B.
<span id="S0.F1.5.2" class="ltx_text ltx_font_smallcaps">ToRA</span> models exhibit remarkable improvements over previous state-of-the-art approaches across all scales. In particular, <span id="S0.F1.6.3" class="ltx_text ltx_font_smallcaps">ToRA</span>-70B notably outperforms GPT-4’s CoT result on MATH and attains comparable results to GPT-4 solving problems with code.
</figcaption>
</figure>
<figure id="S0.F2" class="ltx_figure"><img src="/html/2309.17452/assets/x2.png" id="S0.F2.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="422" height="228" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Examples of three reasoning formats for mathematical reasoning: (a) Rationale-based methods (e.g., CoT prompting) generate step-by-step natural language rationales, (b) Program-based methods (e.g., PAL prompting) solve tasks with program synthesis, and (c) our proposed Tool-integrated Reasoning format interleaves rationales with program-based tool use.
For brevity, we present a simple example of <em id="S0.F2.10.1" class="ltx_emph ltx_font_italic">single-round tool interaction</em>, where the model creates rationale <math id="S0.F2.5.m1.1" class="ltx_Math" alttext="r_{1}" display="inline"><semantics id="S0.F2.5.m1.1b"><msub id="S0.F2.5.m1.1.1" xref="S0.F2.5.m1.1.1.cmml"><mi id="S0.F2.5.m1.1.1.2" xref="S0.F2.5.m1.1.1.2.cmml">r</mi><mn id="S0.F2.5.m1.1.1.3" xref="S0.F2.5.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S0.F2.5.m1.1c"><apply id="S0.F2.5.m1.1.1.cmml" xref="S0.F2.5.m1.1.1"><csymbol cd="ambiguous" id="S0.F2.5.m1.1.1.1.cmml" xref="S0.F2.5.m1.1.1">subscript</csymbol><ci id="S0.F2.5.m1.1.1.2.cmml" xref="S0.F2.5.m1.1.1.2">𝑟</ci><cn type="integer" id="S0.F2.5.m1.1.1.3.cmml" xref="S0.F2.5.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.F2.5.m1.1d">r_{1}</annotation></semantics></math> for analysis, writes program <math id="S0.F2.6.m2.1" class="ltx_Math" alttext="a_{1}" display="inline"><semantics id="S0.F2.6.m2.1b"><msub id="S0.F2.6.m2.1.1" xref="S0.F2.6.m2.1.1.cmml"><mi id="S0.F2.6.m2.1.1.2" xref="S0.F2.6.m2.1.1.2.cmml">a</mi><mn id="S0.F2.6.m2.1.1.3" xref="S0.F2.6.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S0.F2.6.m2.1c"><apply id="S0.F2.6.m2.1.1.cmml" xref="S0.F2.6.m2.1.1"><csymbol cd="ambiguous" id="S0.F2.6.m2.1.1.1.cmml" xref="S0.F2.6.m2.1.1">subscript</csymbol><ci id="S0.F2.6.m2.1.1.2.cmml" xref="S0.F2.6.m2.1.1.2">𝑎</ci><cn type="integer" id="S0.F2.6.m2.1.1.3.cmml" xref="S0.F2.6.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.F2.6.m2.1d">a_{1}</annotation></semantics></math> to call an external solver, obtains the execution output <math id="S0.F2.7.m3.1" class="ltx_Math" alttext="o_{1}" display="inline"><semantics id="S0.F2.7.m3.1b"><msub id="S0.F2.7.m3.1.1" xref="S0.F2.7.m3.1.1.cmml"><mi id="S0.F2.7.m3.1.1.2" xref="S0.F2.7.m3.1.1.2.cmml">o</mi><mn id="S0.F2.7.m3.1.1.3" xref="S0.F2.7.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S0.F2.7.m3.1c"><apply id="S0.F2.7.m3.1.1.cmml" xref="S0.F2.7.m3.1.1"><csymbol cd="ambiguous" id="S0.F2.7.m3.1.1.1.cmml" xref="S0.F2.7.m3.1.1">subscript</csymbol><ci id="S0.F2.7.m3.1.1.2.cmml" xref="S0.F2.7.m3.1.1.2">𝑜</ci><cn type="integer" id="S0.F2.7.m3.1.1.3.cmml" xref="S0.F2.7.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.F2.7.m3.1d">o_{1}</annotation></semantics></math>, and then generates rationale <math id="S0.F2.8.m4.1" class="ltx_Math" alttext="r_{2}" display="inline"><semantics id="S0.F2.8.m4.1b"><msub id="S0.F2.8.m4.1.1" xref="S0.F2.8.m4.1.1.cmml"><mi id="S0.F2.8.m4.1.1.2" xref="S0.F2.8.m4.1.1.2.cmml">r</mi><mn id="S0.F2.8.m4.1.1.3" xref="S0.F2.8.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S0.F2.8.m4.1c"><apply id="S0.F2.8.m4.1.1.cmml" xref="S0.F2.8.m4.1.1"><csymbol cd="ambiguous" id="S0.F2.8.m4.1.1.1.cmml" xref="S0.F2.8.m4.1.1">subscript</csymbol><ci id="S0.F2.8.m4.1.1.2.cmml" xref="S0.F2.8.m4.1.1.2">𝑟</ci><cn type="integer" id="S0.F2.8.m4.1.1.3.cmml" xref="S0.F2.8.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.F2.8.m4.1d">r_{2}</annotation></semantics></math> to finalize the answer.
</figcaption>
</figure>
<section id="S1" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2>

<div id="S1.p1" class="ltx_para ltx_noindent">
<p id="S1.p1.1" class="ltx_p">Large language models (LLMs), such as GPT-4 <cite class="ltx_cite ltx_citemacro_citep">(OpenAI, <a href="#bib.bib29" title="" class="ltx_ref">2023</a>)</cite> and PaLM-2 <cite class="ltx_cite ltx_citemacro_citep">(Anil et al., <a href="#bib.bib1" title="" class="ltx_ref">2023</a>)</cite>, have demonstrated remarkable progress in a wide range of language tasks, particularly in the longstanding challenge of mathematical reasoning <cite class="ltx_cite ltx_citemacro_citep">(Feigenbaum et al., <a href="#bib.bib8" title="" class="ltx_ref">1963</a>; Hosseini et al., <a href="#bib.bib15" title="" class="ltx_ref">2014</a>)</cite>. However, open-source models, such as LLaMA-2 <cite class="ltx_cite ltx_citemacro_citep">(Touvron et al., <a href="#bib.bib43" title="" class="ltx_ref">2023a</a>; <a href="#bib.bib44" title="" class="ltx_ref">b</a>)</cite> and Falcon <cite class="ltx_cite ltx_citemacro_citep">(Penedo et al., <a href="#bib.bib33" title="" class="ltx_ref">2023</a>)</cite>, still struggle with advanced mathematical reasoning tasks.</p>
</div>
<div id="S1.p2" class="ltx_para ltx_noindent">
<p id="S1.p2.1" class="ltx_p">Existing works improve mathematical performance of language models either with step-by-step natural language reasoning <cite class="ltx_cite ltx_citemacro_citep">(Wei et al., <a href="#bib.bib45" title="" class="ltx_ref">2022</a>)</cite> as illustrated in Fig <a href="#S0.F2" title="Figure 2 ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a> (a), or by synthesizing and executing programs to obtain the answers <cite class="ltx_cite ltx_citemacro_citep">(Gao et al., <a href="#bib.bib10" title="" class="ltx_ref">2022</a>; Chen et al., <a href="#bib.bib5" title="" class="ltx_ref">2022</a>)</cite>, as depicted in Fig <a href="#S0.F2" title="Figure 2 ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a> (b).
Both approaches exhibit complementary advantages.
Natural language is suitable for semantic analysis, planning, and abstract reasoning (e.g., commonsense reasoning), but struggles with precise computation, symbolic manipulation, and algorithmic processing.
Conversely, programs excel in rigorous operations, and can outsource intricate calculations to specialized tools like equation solvers.</p>
</div>
<div id="S1.p3" class="ltx_para ltx_noindent">
<p id="S1.p3.1" class="ltx_p">To leverage the benefits of both natural language reasoning and program-based tool use, we train open-source models such as LLaMA-2 to reason in a way where natural language reasoning is interleaved with program-based tool use synergistically (as depicted in Fig <a href="#S0.F2" title="Figure 2 ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a> (c)), thereby largely reducing the gap with closed-source models like GPT-4 in mathematical reasoning.
Specifically, we first design the interleaving format of reasoning, curate corresponding interactive tool-use trajectories for mathematical problems from the popular GSM8k <cite class="ltx_cite ltx_citemacro_citep">(Cobbe et al., <a href="#bib.bib6" title="" class="ltx_ref">2021</a>)</cite> and MATH <cite class="ltx_cite ltx_citemacro_citep">(Hendrycks et al., <a href="#bib.bib12" title="" class="ltx_ref">2021</a>)</cite> dataset, and then apply imitation learning on the high-quality annotations, leading to a better performance than any existing open-source model.
Furthermore, since the curated data is far from exhausting all valid trajectories for a problem, relying solely on imitation learning restricts a model’s output space, hindering the flexibility in exploring plausible trajectories during testing.
To improve the diversity of plausible reasoning steps and mitigate improper tool-use behavior, we apply <em id="S1.p3.1.1" class="ltx_emph ltx_font_italic">output space shaping</em> which additionally trains the models on both self-sampled valid trajectories and invalid ones that have been corrected by a teacher model (e.g., a 34B model can serve as the teacher for a 7B model).
<em id="S1.p3.1.2" class="ltx_emph ltx_font_italic">Output space shaping</em> significantly boosts reasoning, allowing open-source models to attain an accuracy exceeding 50% on the competition-level MATH dataset for the first time.</p>
</div>
<div id="S1.p4" class="ltx_para ltx_noindent">
<p id="S1.p4.1" class="ltx_p">We evaluate the resulting suite of <span id="S1.p4.1.1" class="ltx_text ltx_framed ltx_framed_underline">To</span>ol-integrated <span id="S1.p4.1.2" class="ltx_text ltx_framed ltx_framed_underline">R</span>easoning <span id="S1.p4.1.3" class="ltx_text ltx_framed ltx_framed_underline">A</span>gents (<span id="S1.p4.1.4" class="ltx_text ltx_font_smallcaps">ToRA</span>) ranging from 7B to 70B on 10 diverse mathematical reasoning datasets.
As shown in Fig <a href="#S0.F1" title="Figure 1 ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>, <span id="S1.p4.1.5" class="ltx_text ltx_font_smallcaps">ToRA</span> series significantly outperform open-source models across all scales.
Notably, on the competition-level MATH dataset, <span id="S1.p4.1.6" class="ltx_text ltx_font_smallcaps">ToRA</span>-7B outperforms the previous SoTA WizardMath-70B <cite class="ltx_cite ltx_citemacro_citep">(Luo et al., <a href="#bib.bib24" title="" class="ltx_ref">2023</a>)</cite> by 22% absolute.
<span id="S1.p4.1.7" class="ltx_text ltx_font_smallcaps">ToRA-Code</span>-34B beats GPT-4’s CoT result <cite class="ltx_cite ltx_citemacro_citep">(Bubeck et al., <a href="#bib.bib3" title="" class="ltx_ref">2023</a>)</cite> by 8.3% absolute (50.8% vs. 42.5%), and is competitive with GPT-4 solving problems with code (GPT-4-Code, 51.8%).
In addition, we analyze the benefits and remaining challenges of tool interaction for mathematical reasoning, providing valuable insights for future work.</p>
</div>
</section>
<section id="S2" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">2 </span><span id="S2.1.1" class="ltx_text ltx_font_smallcaps">ToRA</span>: Tool-Integrated Agents for Mathematical Reasoning</h2>

<figure id="S2.F3" class="ltx_figure"><img src="/html/2309.17452/assets/x3.png" id="S2.F3.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="422" height="171" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>
Training <span id="S2.F3.15.1" class="ltx_text ltx_font_smallcaps">ToRA</span> contains two steps.
① <span id="S2.F3.16.2" class="ltx_text ltx_font_bold">Imitation Learning</span>: Prompt LLMs like GPT-4 to generate Tool-integrated Reasoning trajectories (<span id="S2.F3.17.3" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span>) and use this corpus to fine-tune a model <math id="S2.F3.5.m1.1" class="ltx_Math" alttext="\mathcal{M}" display="inline"><semantics id="S2.F3.5.m1.1b"><mi class="ltx_font_mathcaligraphic" id="S2.F3.5.m1.1.1" xref="S2.F3.5.m1.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S2.F3.5.m1.1c"><ci id="S2.F3.5.m1.1.1.cmml" xref="S2.F3.5.m1.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F3.5.m1.1d">\mathcal{M}</annotation></semantics></math>;
② <span id="S2.F3.18.4" class="ltx_text ltx_font_bold">Output Space Shaping</span>: Sample diverse tool-use trajectories with <math id="S2.F3.6.m2.1" class="ltx_Math" alttext="\mathcal{M}" display="inline"><semantics id="S2.F3.6.m2.1b"><mi class="ltx_font_mathcaligraphic" id="S2.F3.6.m2.1.1" xref="S2.F3.6.m2.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S2.F3.6.m2.1c"><ci id="S2.F3.6.m2.1.1.cmml" xref="S2.F3.6.m2.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F3.6.m2.1d">\mathcal{M}</annotation></semantics></math>, keep the valid ones, correct the invalid ones with a teacher model <math id="S2.F3.7.m3.1" class="ltx_Math" alttext="\mathcal{M}^{\prime}" display="inline"><semantics id="S2.F3.7.m3.1b"><msup id="S2.F3.7.m3.1.1" xref="S2.F3.7.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.F3.7.m3.1.1.2" xref="S2.F3.7.m3.1.1.2.cmml">ℳ</mi><mo id="S2.F3.7.m3.1.1.3" xref="S2.F3.7.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.F3.7.m3.1c"><apply id="S2.F3.7.m3.1.1.cmml" xref="S2.F3.7.m3.1.1"><csymbol cd="ambiguous" id="S2.F3.7.m3.1.1.1.cmml" xref="S2.F3.7.m3.1.1">superscript</csymbol><ci id="S2.F3.7.m3.1.1.2.cmml" xref="S2.F3.7.m3.1.1.2">ℳ</ci><ci id="S2.F3.7.m3.1.1.3.cmml" xref="S2.F3.7.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F3.7.m3.1d">\mathcal{M}^{\prime}</annotation></semantics></math>, and retrain <math id="S2.F3.8.m4.1" class="ltx_Math" alttext="\mathcal{M}" display="inline"><semantics id="S2.F3.8.m4.1b"><mi class="ltx_font_mathcaligraphic" id="S2.F3.8.m4.1.1" xref="S2.F3.8.m4.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S2.F3.8.m4.1c"><ci id="S2.F3.8.m4.1.1.cmml" xref="S2.F3.8.m4.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F3.8.m4.1d">\mathcal{M}</annotation></semantics></math> on the union of sampled valid trajectories, corrected ones, and the initial <span id="S2.F3.19.5" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span> to obtain <span id="S2.F3.20.6" class="ltx_text ltx_font_smallcaps">ToRA</span>.
</figcaption>
</figure>
<section id="S2.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">2.1 </span>Overview</h3>

<div id="S2.SS1.p1" class="ltx_para ltx_noindent">
<p id="S2.SS1.p1.7" class="ltx_p"><span id="S2.SS1.p1.7.1" class="ltx_text ltx_font_smallcaps">ToRA</span> series solve challenging mathematical problems by leveraging both natural language reasoning and program-based tool use.
As shown in Fig <a href="#S0.F2" title="Figure 2 ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a> (c), given a mathematical problem <math id="S2.SS1.p1.1.m1.1" class="ltx_Math" alttext="q" display="inline"><semantics id="S2.SS1.p1.1.m1.1a"><mi id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.1b"><ci id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.1c">q</annotation></semantics></math>, <span id="S2.SS1.p1.7.2" class="ltx_text ltx_font_smallcaps">ToRA</span> reasons with natural language, producing <math id="S2.SS1.p1.2.m2.1" class="ltx_Math" alttext="r_{1}" display="inline"><semantics id="S2.SS1.p1.2.m2.1a"><msub id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml"><mi id="S2.SS1.p1.2.m2.1.1.2" xref="S2.SS1.p1.2.m2.1.1.2.cmml">r</mi><mn id="S2.SS1.p1.2.m2.1.1.3" xref="S2.SS1.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.2.m2.1b"><apply id="S2.SS1.p1.2.m2.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.2.m2.1.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1">subscript</csymbol><ci id="S2.SS1.p1.2.m2.1.1.2.cmml" xref="S2.SS1.p1.2.m2.1.1.2">𝑟</ci><cn type="integer" id="S2.SS1.p1.2.m2.1.1.3.cmml" xref="S2.SS1.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.2.m2.1c">r_{1}</annotation></semantics></math>.
When reaching a point where program-based tool use is more appropriate for the subsequent task, e.g., equation solving, <span id="S2.SS1.p1.7.3" class="ltx_text ltx_font_smallcaps">ToRA</span> generates a program <math id="S2.SS1.p1.3.m3.1" class="ltx_Math" alttext="a_{1}" display="inline"><semantics id="S2.SS1.p1.3.m3.1a"><msub id="S2.SS1.p1.3.m3.1.1" xref="S2.SS1.p1.3.m3.1.1.cmml"><mi id="S2.SS1.p1.3.m3.1.1.2" xref="S2.SS1.p1.3.m3.1.1.2.cmml">a</mi><mn id="S2.SS1.p1.3.m3.1.1.3" xref="S2.SS1.p1.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.3.m3.1b"><apply id="S2.SS1.p1.3.m3.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.3.m3.1.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1">subscript</csymbol><ci id="S2.SS1.p1.3.m3.1.1.2.cmml" xref="S2.SS1.p1.3.m3.1.1.2">𝑎</ci><cn type="integer" id="S2.SS1.p1.3.m3.1.1.3.cmml" xref="S2.SS1.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m3.1c">a_{1}</annotation></semantics></math> for tool use following natural language guidance <math id="S2.SS1.p1.4.m4.1" class="ltx_Math" alttext="r_{1}" display="inline"><semantics id="S2.SS1.p1.4.m4.1a"><msub id="S2.SS1.p1.4.m4.1.1" xref="S2.SS1.p1.4.m4.1.1.cmml"><mi id="S2.SS1.p1.4.m4.1.1.2" xref="S2.SS1.p1.4.m4.1.1.2.cmml">r</mi><mn id="S2.SS1.p1.4.m4.1.1.3" xref="S2.SS1.p1.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m4.1b"><apply id="S2.SS1.p1.4.m4.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.4.m4.1.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1">subscript</csymbol><ci id="S2.SS1.p1.4.m4.1.1.2.cmml" xref="S2.SS1.p1.4.m4.1.1.2">𝑟</ci><cn type="integer" id="S2.SS1.p1.4.m4.1.1.3.cmml" xref="S2.SS1.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m4.1c">r_{1}</annotation></semantics></math>.
The execution output <math id="S2.SS1.p1.5.m5.1" class="ltx_Math" alttext="o_{1}" display="inline"><semantics id="S2.SS1.p1.5.m5.1a"><msub id="S2.SS1.p1.5.m5.1.1" xref="S2.SS1.p1.5.m5.1.1.cmml"><mi id="S2.SS1.p1.5.m5.1.1.2" xref="S2.SS1.p1.5.m5.1.1.2.cmml">o</mi><mn id="S2.SS1.p1.5.m5.1.1.3" xref="S2.SS1.p1.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.5.m5.1b"><apply id="S2.SS1.p1.5.m5.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.5.m5.1.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1">subscript</csymbol><ci id="S2.SS1.p1.5.m5.1.1.2.cmml" xref="S2.SS1.p1.5.m5.1.1.2">𝑜</ci><cn type="integer" id="S2.SS1.p1.5.m5.1.1.3.cmml" xref="S2.SS1.p1.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.5.m5.1c">o_{1}</annotation></semantics></math> will be fed to <span id="S2.SS1.p1.7.4" class="ltx_text ltx_font_smallcaps">ToRA</span> for subsequent processing including tool use adjustments, sub-tasks solving, or answer finalization.
We repeat the process until the model places its answer within “<span id="S2.SS1.p1.7.5" class="ltx_text ltx_lstlisting ltx_font_typewriter ltx_framed ltx_framed_topbottom">\<span id="S2.SS1.p1.7.5.1" class="ltx_text ltx_lst_identifier">boxed</span>{}</span>”.
The resulting trajectory is denoted as <math id="S2.SS1.p1.6.m6.1" class="ltx_Math" alttext="\tau=r_{1}a_{1}o_{1}...r_{n-1}a_{n-1}o_{n-1}r_{n}" display="inline"><semantics id="S2.SS1.p1.6.m6.1a"><mrow id="S2.SS1.p1.6.m6.1.1" xref="S2.SS1.p1.6.m6.1.1.cmml"><mi id="S2.SS1.p1.6.m6.1.1.2" xref="S2.SS1.p1.6.m6.1.1.2.cmml">τ</mi><mo id="S2.SS1.p1.6.m6.1.1.1" xref="S2.SS1.p1.6.m6.1.1.1.cmml">=</mo><mrow id="S2.SS1.p1.6.m6.1.1.3" xref="S2.SS1.p1.6.m6.1.1.3.cmml"><msub id="S2.SS1.p1.6.m6.1.1.3.2" xref="S2.SS1.p1.6.m6.1.1.3.2.cmml"><mi id="S2.SS1.p1.6.m6.1.1.3.2.2" xref="S2.SS1.p1.6.m6.1.1.3.2.2.cmml">r</mi><mn id="S2.SS1.p1.6.m6.1.1.3.2.3" xref="S2.SS1.p1.6.m6.1.1.3.2.3.cmml">1</mn></msub><mo lspace="0em" rspace="0em" id="S2.SS1.p1.6.m6.1.1.3.1" xref="S2.SS1.p1.6.m6.1.1.3.1.cmml">​</mo><msub id="S2.SS1.p1.6.m6.1.1.3.3" xref="S2.SS1.p1.6.m6.1.1.3.3.cmml"><mi id="S2.SS1.p1.6.m6.1.1.3.3.2" xref="S2.SS1.p1.6.m6.1.1.3.3.2.cmml">a</mi><mn id="S2.SS1.p1.6.m6.1.1.3.3.3" xref="S2.SS1.p1.6.m6.1.1.3.3.3.cmml">1</mn></msub><mo lspace="0em" rspace="0em" id="S2.SS1.p1.6.m6.1.1.3.1a" xref="S2.SS1.p1.6.m6.1.1.3.1.cmml">​</mo><msub id="S2.SS1.p1.6.m6.1.1.3.4" xref="S2.SS1.p1.6.m6.1.1.3.4.cmml"><mi id="S2.SS1.p1.6.m6.1.1.3.4.2" xref="S2.SS1.p1.6.m6.1.1.3.4.2.cmml">o</mi><mn id="S2.SS1.p1.6.m6.1.1.3.4.3" xref="S2.SS1.p1.6.m6.1.1.3.4.3.cmml">1</mn></msub><mo lspace="0em" rspace="0em" id="S2.SS1.p1.6.m6.1.1.3.1b" xref="S2.SS1.p1.6.m6.1.1.3.1.cmml">​</mo><mi mathvariant="normal" id="S2.SS1.p1.6.m6.1.1.3.5" xref="S2.SS1.p1.6.m6.1.1.3.5.cmml">…</mi><mo lspace="0em" rspace="0em" id="S2.SS1.p1.6.m6.1.1.3.1c" xref="S2.SS1.p1.6.m6.1.1.3.1.cmml">​</mo><msub id="S2.SS1.p1.6.m6.1.1.3.6" xref="S2.SS1.p1.6.m6.1.1.3.6.cmml"><mi id="S2.SS1.p1.6.m6.1.1.3.6.2" xref="S2.SS1.p1.6.m6.1.1.3.6.2.cmml">r</mi><mrow id="S2.SS1.p1.6.m6.1.1.3.6.3" xref="S2.SS1.p1.6.m6.1.1.3.6.3.cmml"><mi id="S2.SS1.p1.6.m6.1.1.3.6.3.2" xref="S2.SS1.p1.6.m6.1.1.3.6.3.2.cmml">n</mi><mo 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xref="S2.SS1.p1.6.m6.1.1.3"><times id="S2.SS1.p1.6.m6.1.1.3.1.cmml" xref="S2.SS1.p1.6.m6.1.1.3.1"></times><apply id="S2.SS1.p1.6.m6.1.1.3.2.cmml" xref="S2.SS1.p1.6.m6.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.1.1.3.2.1.cmml" xref="S2.SS1.p1.6.m6.1.1.3.2">subscript</csymbol><ci id="S2.SS1.p1.6.m6.1.1.3.2.2.cmml" xref="S2.SS1.p1.6.m6.1.1.3.2.2">𝑟</ci><cn type="integer" id="S2.SS1.p1.6.m6.1.1.3.2.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3.2.3">1</cn></apply><apply id="S2.SS1.p1.6.m6.1.1.3.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.1.1.3.3.1.cmml" xref="S2.SS1.p1.6.m6.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p1.6.m6.1.1.3.3.2.cmml" xref="S2.SS1.p1.6.m6.1.1.3.3.2">𝑎</ci><cn type="integer" id="S2.SS1.p1.6.m6.1.1.3.3.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3.3.3">1</cn></apply><apply id="S2.SS1.p1.6.m6.1.1.3.4.cmml" xref="S2.SS1.p1.6.m6.1.1.3.4"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.1.1.3.4.1.cmml" xref="S2.SS1.p1.6.m6.1.1.3.4">subscript</csymbol><ci id="S2.SS1.p1.6.m6.1.1.3.4.2.cmml" xref="S2.SS1.p1.6.m6.1.1.3.4.2">𝑜</ci><cn type="integer" id="S2.SS1.p1.6.m6.1.1.3.4.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3.4.3">1</cn></apply><ci id="S2.SS1.p1.6.m6.1.1.3.5.cmml" xref="S2.SS1.p1.6.m6.1.1.3.5">…</ci><apply id="S2.SS1.p1.6.m6.1.1.3.6.cmml" xref="S2.SS1.p1.6.m6.1.1.3.6"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.1.1.3.6.1.cmml" xref="S2.SS1.p1.6.m6.1.1.3.6">subscript</csymbol><ci id="S2.SS1.p1.6.m6.1.1.3.6.2.cmml" xref="S2.SS1.p1.6.m6.1.1.3.6.2">𝑟</ci><apply id="S2.SS1.p1.6.m6.1.1.3.6.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3.6.3"><minus id="S2.SS1.p1.6.m6.1.1.3.6.3.1.cmml" xref="S2.SS1.p1.6.m6.1.1.3.6.3.1"></minus><ci id="S2.SS1.p1.6.m6.1.1.3.6.3.2.cmml" xref="S2.SS1.p1.6.m6.1.1.3.6.3.2">𝑛</ci><cn type="integer" id="S2.SS1.p1.6.m6.1.1.3.6.3.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3.6.3.3">1</cn></apply></apply><apply id="S2.SS1.p1.6.m6.1.1.3.7.cmml" xref="S2.SS1.p1.6.m6.1.1.3.7"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.1.1.3.7.1.cmml" xref="S2.SS1.p1.6.m6.1.1.3.7">subscript</csymbol><ci id="S2.SS1.p1.6.m6.1.1.3.7.2.cmml" xref="S2.SS1.p1.6.m6.1.1.3.7.2">𝑎</ci><apply id="S2.SS1.p1.6.m6.1.1.3.7.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3.7.3"><minus id="S2.SS1.p1.6.m6.1.1.3.7.3.1.cmml" xref="S2.SS1.p1.6.m6.1.1.3.7.3.1"></minus><ci id="S2.SS1.p1.6.m6.1.1.3.7.3.2.cmml" xref="S2.SS1.p1.6.m6.1.1.3.7.3.2">𝑛</ci><cn type="integer" id="S2.SS1.p1.6.m6.1.1.3.7.3.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3.7.3.3">1</cn></apply></apply><apply id="S2.SS1.p1.6.m6.1.1.3.8.cmml" xref="S2.SS1.p1.6.m6.1.1.3.8"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.1.1.3.8.1.cmml" xref="S2.SS1.p1.6.m6.1.1.3.8">subscript</csymbol><ci id="S2.SS1.p1.6.m6.1.1.3.8.2.cmml" xref="S2.SS1.p1.6.m6.1.1.3.8.2">𝑜</ci><apply id="S2.SS1.p1.6.m6.1.1.3.8.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3.8.3"><minus id="S2.SS1.p1.6.m6.1.1.3.8.3.1.cmml" xref="S2.SS1.p1.6.m6.1.1.3.8.3.1"></minus><ci id="S2.SS1.p1.6.m6.1.1.3.8.3.2.cmml" xref="S2.SS1.p1.6.m6.1.1.3.8.3.2">𝑛</ci><cn type="integer" id="S2.SS1.p1.6.m6.1.1.3.8.3.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3.8.3.3">1</cn></apply></apply><apply id="S2.SS1.p1.6.m6.1.1.3.9.cmml" xref="S2.SS1.p1.6.m6.1.1.3.9"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.1.1.3.9.1.cmml" xref="S2.SS1.p1.6.m6.1.1.3.9">subscript</csymbol><ci id="S2.SS1.p1.6.m6.1.1.3.9.2.cmml" xref="S2.SS1.p1.6.m6.1.1.3.9.2">𝑟</ci><ci id="S2.SS1.p1.6.m6.1.1.3.9.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3.9.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.6.m6.1c">\tau=r_{1}a_{1}o_{1}...r_{n-1}a_{n-1}o_{n-1}r_{n}</annotation></semantics></math>, where <math id="S2.SS1.p1.7.m7.1" class="ltx_Math" alttext="r_{n}" display="inline"><semantics id="S2.SS1.p1.7.m7.1a"><msub id="S2.SS1.p1.7.m7.1.1" xref="S2.SS1.p1.7.m7.1.1.cmml"><mi id="S2.SS1.p1.7.m7.1.1.2" xref="S2.SS1.p1.7.m7.1.1.2.cmml">r</mi><mi id="S2.SS1.p1.7.m7.1.1.3" xref="S2.SS1.p1.7.m7.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.7.m7.1b"><apply id="S2.SS1.p1.7.m7.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.7.m7.1.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1">subscript</csymbol><ci id="S2.SS1.p1.7.m7.1.1.2.cmml" xref="S2.SS1.p1.7.m7.1.1.2">𝑟</ci><ci id="S2.SS1.p1.7.m7.1.1.3.cmml" xref="S2.SS1.p1.7.m7.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.7.m7.1c">r_{n}</annotation></semantics></math> contains the answer.</p>
</div>
<div id="S2.SS1.p2" class="ltx_para ltx_noindent">
<p id="S2.SS1.p2.1" class="ltx_p">Fig <a href="#S2.F3" title="Figure 3 ‣ 2 ToRA: Tool-Integrated Agents for Mathematical Reasoning ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3</span></a> presents the training pipeline of <span id="S2.SS1.p2.1.1" class="ltx_text ltx_font_smallcaps">ToRA</span>.
We first collect interactive tool-use trajectories on popular mathematical datasets.
We then apply imitation learning on the resulting annotations, as well as output space shaping to further refine models’ reasoning behavior.</p>
</div>
</section>
<section id="S2.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">2.2 </span>Collecting Interactive Tool-Use Trajectories</h3>

<figure id="S2.SS2.fig1" class="ltx_figure">
<figure id="alg1" class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top">
<div id="alg1.2" class="ltx_listing ltx_listing">
<div id="alg0.l1" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline">1:</span>problem <math id="alg0.l1.m1.1" class="ltx_Math" alttext="q" display="inline"><semantics id="alg0.l1.m1.1a"><mi id="alg0.l1.m1.1.1" xref="alg0.l1.m1.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="alg0.l1.m1.1b"><ci id="alg0.l1.m1.1.1.cmml" xref="alg0.l1.m1.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="alg0.l1.m1.1c">q</annotation></semantics></math>, model <math id="alg0.l1.m2.1" class="ltx_Math" alttext="\mathcal{G}" display="inline"><semantics id="alg0.l1.m2.1a"><mi class="ltx_font_mathcaligraphic" id="alg0.l1.m2.1.1" xref="alg0.l1.m2.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="alg0.l1.m2.1b"><ci id="alg0.l1.m2.1.1.cmml" xref="alg0.l1.m2.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="alg0.l1.m2.1c">\mathcal{G}</annotation></semantics></math>, prompt <math id="alg0.l1.m3.1" class="ltx_Math" alttext="{p}" display="inline"><semantics id="alg0.l1.m3.1a"><mi id="alg0.l1.m3.1.1" xref="alg0.l1.m3.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="alg0.l1.m3.1b"><ci id="alg0.l1.m3.1.1.cmml" xref="alg0.l1.m3.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="alg0.l1.m3.1c">{p}</annotation></semantics></math>, external tools <math id="alg0.l1.m4.1" class="ltx_Math" alttext="\mathcal{E}" display="inline"><semantics id="alg0.l1.m4.1a"><mi class="ltx_font_mathcaligraphic" id="alg0.l1.m4.1.1" xref="alg0.l1.m4.1.1.cmml">ℰ</mi><annotation-xml encoding="MathML-Content" id="alg0.l1.m4.1b"><ci id="alg0.l1.m4.1.1.cmml" xref="alg0.l1.m4.1.1">ℰ</ci></annotation-xml><annotation encoding="application/x-tex" id="alg0.l1.m4.1c">\mathcal{E}</annotation></semantics></math>, stop condition <span id="alg0.l1.1" class="ltx_text ltx_font_italic">Stop(<math id="alg0.l1.1.m1.1" class="ltx_Math" alttext="\cdot" display="inline"><semantics id="alg0.l1.1.m1.1a"><mo id="alg0.l1.1.m1.1.1" xref="alg0.l1.1.m1.1.1.cmml">⋅</mo><annotation-xml encoding="MathML-Content" id="alg0.l1.1.m1.1b"><ci id="alg0.l1.1.m1.1.1.cmml" xref="alg0.l1.1.m1.1.1">⋅</ci></annotation-xml><annotation encoding="application/x-tex" id="alg0.l1.1.m1.1c">\cdot</annotation></semantics></math>)</span>, maximum iteration rounds <math id="alg0.l1.m5.1" class="ltx_Math" alttext="n" display="inline"><semantics id="alg0.l1.m5.1a"><mi id="alg0.l1.m5.1.1" xref="alg0.l1.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="alg0.l1.m5.1b"><ci id="alg0.l1.m5.1.1.cmml" xref="alg0.l1.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="alg0.l1.m5.1c">n</annotation></semantics></math>

</div>
<div id="alg0.l2" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline">2:</span><math id="alg0.l2.m1.1" class="ltx_Math" alttext='\tau_{0}\leftarrow\text{""}' display="inline"><semantics id="alg0.l2.m1.1a"><mrow id="alg0.l2.m1.1.1" xref="alg0.l2.m1.1.1.cmml"><msub id="alg0.l2.m1.1.1.2" xref="alg0.l2.m1.1.1.2.cmml"><mi id="alg0.l2.m1.1.1.2.2" xref="alg0.l2.m1.1.1.2.2.cmml">τ</mi><mn id="alg0.l2.m1.1.1.2.3" xref="alg0.l2.m1.1.1.2.3.cmml">0</mn></msub><mo stretchy="false" id="alg0.l2.m1.1.1.1" xref="alg0.l2.m1.1.1.1.cmml">←</mo><mtext id="alg0.l2.m1.1.1.3" xref="alg0.l2.m1.1.1.3a.cmml">""</mtext></mrow><annotation-xml encoding="MathML-Content" id="alg0.l2.m1.1b"><apply id="alg0.l2.m1.1.1.cmml" xref="alg0.l2.m1.1.1"><ci id="alg0.l2.m1.1.1.1.cmml" xref="alg0.l2.m1.1.1.1">←</ci><apply id="alg0.l2.m1.1.1.2.cmml" xref="alg0.l2.m1.1.1.2"><csymbol cd="ambiguous" id="alg0.l2.m1.1.1.2.1.cmml" xref="alg0.l2.m1.1.1.2">subscript</csymbol><ci id="alg0.l2.m1.1.1.2.2.cmml" xref="alg0.l2.m1.1.1.2.2">𝜏</ci><cn type="integer" id="alg0.l2.m1.1.1.2.3.cmml" xref="alg0.l2.m1.1.1.2.3">0</cn></apply><ci id="alg0.l2.m1.1.1.3a.cmml" xref="alg0.l2.m1.1.1.3"><mtext id="alg0.l2.m1.1.1.3.cmml" xref="alg0.l2.m1.1.1.3">""</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg0.l2.m1.1c">\tau_{0}\leftarrow\text{""}</annotation></semantics></math> <span id="alg0.l2.1" class="ltx_text" style="float:right;"><math id="alg0.l2.1.m1.1" class="ltx_Math" alttext="\triangleright" display="inline"><semantics id="alg0.l2.1.m1.1a"><mo id="alg0.l2.1.m1.1.1" xref="alg0.l2.1.m1.1.1.cmml">▷</mo><annotation-xml encoding="MathML-Content" id="alg0.l2.1.m1.1b"><ci id="alg0.l2.1.m1.1.1.cmml" xref="alg0.l2.1.m1.1.1">▷</ci></annotation-xml><annotation encoding="application/x-tex" id="alg0.l2.1.m1.1c">\triangleright</annotation></semantics></math> Trajectory Initialization
</span>
</div>
<div id="alg0.l3" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline">3:</span><span id="alg0.l3.1" class="ltx_text ltx_font_bold">for</span> <math id="alg0.l3.m1.1" class="ltx_Math" alttext="i\leftarrow 1" display="inline"><semantics id="alg0.l3.m1.1a"><mrow id="alg0.l3.m1.1.1" xref="alg0.l3.m1.1.1.cmml"><mi id="alg0.l3.m1.1.1.2" xref="alg0.l3.m1.1.1.2.cmml">i</mi><mo stretchy="false" id="alg0.l3.m1.1.1.1" xref="alg0.l3.m1.1.1.1.cmml">←</mo><mn id="alg0.l3.m1.1.1.3" xref="alg0.l3.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="alg0.l3.m1.1b"><apply id="alg0.l3.m1.1.1.cmml" xref="alg0.l3.m1.1.1"><ci id="alg0.l3.m1.1.1.1.cmml" xref="alg0.l3.m1.1.1.1">←</ci><ci id="alg0.l3.m1.1.1.2.cmml" xref="alg0.l3.m1.1.1.2">𝑖</ci><cn type="integer" id="alg0.l3.m1.1.1.3.cmml" xref="alg0.l3.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg0.l3.m1.1c">i\leftarrow 1</annotation></semantics></math> to <math id="alg0.l3.m2.1" class="ltx_Math" alttext="n" display="inline"><semantics id="alg0.l3.m2.1a"><mi id="alg0.l3.m2.1.1" xref="alg0.l3.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="alg0.l3.m2.1b"><ci id="alg0.l3.m2.1.1.cmml" xref="alg0.l3.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="alg0.l3.m2.1c">n</annotation></semantics></math> <span id="alg0.l3.2" class="ltx_text ltx_font_bold">do</span>

</div>
<div id="alg0.l4" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline">4:</span>     <math id="alg0.l4.m1.1" class="ltx_math_unparsed" alttext="r_{i}\sim\mathbb{P}_{\mathcal{G}}(\cdot|p\oplus q\oplus\tau_{i-1})" display="inline"><semantics id="alg0.l4.m1.1a"><mrow id="alg0.l4.m1.1b"><msub id="alg0.l4.m1.1.1"><mi id="alg0.l4.m1.1.1.2">r</mi><mi id="alg0.l4.m1.1.1.3">i</mi></msub><mo id="alg0.l4.m1.1.2">∼</mo><msub id="alg0.l4.m1.1.3"><mi id="alg0.l4.m1.1.3.2">ℙ</mi><mi class="ltx_font_mathcaligraphic" id="alg0.l4.m1.1.3.3">𝒢</mi></msub><mrow id="alg0.l4.m1.1.4"><mo stretchy="false" id="alg0.l4.m1.1.4.1">(</mo><mo lspace="0em" rspace="0em" id="alg0.l4.m1.1.4.2">⋅</mo><mo fence="false" rspace="0.167em" stretchy="false" id="alg0.l4.m1.1.4.3">|</mo><mi id="alg0.l4.m1.1.4.4">p</mi><mo id="alg0.l4.m1.1.4.5">⊕</mo><mi id="alg0.l4.m1.1.4.6">q</mi><mo id="alg0.l4.m1.1.4.7">⊕</mo><msub id="alg0.l4.m1.1.4.8"><mi id="alg0.l4.m1.1.4.8.2">τ</mi><mrow id="alg0.l4.m1.1.4.8.3"><mi id="alg0.l4.m1.1.4.8.3.2">i</mi><mo id="alg0.l4.m1.1.4.8.3.1">−</mo><mn id="alg0.l4.m1.1.4.8.3.3">1</mn></mrow></msub><mo stretchy="false" id="alg0.l4.m1.1.4.9">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="alg0.l4.m1.1c">r_{i}\sim\mathbb{P}_{\mathcal{G}}(\cdot|p\oplus q\oplus\tau_{i-1})</annotation></semantics></math> <span id="alg0.l4.1" class="ltx_text" style="float:right;"><math id="alg0.l4.1.m1.1" class="ltx_Math" alttext="\triangleright" display="inline"><semantics id="alg0.l4.1.m1.1a"><mo id="alg0.l4.1.m1.1.1" xref="alg0.l4.1.m1.1.1.cmml">▷</mo><annotation-xml encoding="MathML-Content" id="alg0.l4.1.m1.1b"><ci id="alg0.l4.1.m1.1.1.cmml" xref="alg0.l4.1.m1.1.1">▷</ci></annotation-xml><annotation encoding="application/x-tex" id="alg0.l4.1.m1.1c">\triangleright</annotation></semantics></math> Rationale Generation (Eq. <a href="#S2.E1" title="In Inference Procedure ‣ 2.2 Collecting Interactive Tool-Use Trajectories ‣ 2 ToRA: Tool-Integrated Agents for Mathematical Reasoning ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>)
</span>
</div>
<div id="alg0.l5" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline">5:</span>     <span id="alg0.l5.2" class="ltx_text ltx_font_bold">if</span> <span id="alg0.l5.3" class="ltx_text ltx_font_italic">Stop</span>(<math id="alg0.l5.m1.1" class="ltx_Math" alttext="r_{i}" display="inline"><semantics id="alg0.l5.m1.1a"><msub id="alg0.l5.m1.1.1" xref="alg0.l5.m1.1.1.cmml"><mi id="alg0.l5.m1.1.1.2" xref="alg0.l5.m1.1.1.2.cmml">r</mi><mi id="alg0.l5.m1.1.1.3" xref="alg0.l5.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="alg0.l5.m1.1b"><apply id="alg0.l5.m1.1.1.cmml" xref="alg0.l5.m1.1.1"><csymbol cd="ambiguous" id="alg0.l5.m1.1.1.1.cmml" xref="alg0.l5.m1.1.1">subscript</csymbol><ci id="alg0.l5.m1.1.1.2.cmml" xref="alg0.l5.m1.1.1.2">𝑟</ci><ci id="alg0.l5.m1.1.1.3.cmml" xref="alg0.l5.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg0.l5.m1.1c">r_{i}</annotation></semantics></math>) <span id="alg0.l5.4" class="ltx_text ltx_font_bold">then</span> <span id="alg0.l5.1" class="ltx_text" style="float:right;"><math id="alg0.l5.1.m1.1" class="ltx_Math" alttext="\triangleright" display="inline"><semantics id="alg0.l5.1.m1.1a"><mo id="alg0.l5.1.m1.1.1" xref="alg0.l5.1.m1.1.1.cmml">▷</mo><annotation-xml encoding="MathML-Content" id="alg0.l5.1.m1.1b"><ci id="alg0.l5.1.m1.1.1.cmml" xref="alg0.l5.1.m1.1.1">▷</ci></annotation-xml><annotation encoding="application/x-tex" id="alg0.l5.1.m1.1c">\triangleright</annotation></semantics></math> Stopping Criteria
</span>
</div>
<div id="alg0.l6" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline">6:</span>         <span id="alg0.l6.1" class="ltx_text ltx_font_bold">return</span> <math id="alg0.l6.m1.1" class="ltx_Math" alttext="\tau_{i-1}\oplus r_{i}" display="inline"><semantics id="alg0.l6.m1.1a"><mrow id="alg0.l6.m1.1.1" xref="alg0.l6.m1.1.1.cmml"><msub id="alg0.l6.m1.1.1.2" xref="alg0.l6.m1.1.1.2.cmml"><mi id="alg0.l6.m1.1.1.2.2" xref="alg0.l6.m1.1.1.2.2.cmml">τ</mi><mrow id="alg0.l6.m1.1.1.2.3" xref="alg0.l6.m1.1.1.2.3.cmml"><mi id="alg0.l6.m1.1.1.2.3.2" xref="alg0.l6.m1.1.1.2.3.2.cmml">i</mi><mo id="alg0.l6.m1.1.1.2.3.1" xref="alg0.l6.m1.1.1.2.3.1.cmml">−</mo><mn id="alg0.l6.m1.1.1.2.3.3" xref="alg0.l6.m1.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="alg0.l6.m1.1.1.1" xref="alg0.l6.m1.1.1.1.cmml">⊕</mo><msub id="alg0.l6.m1.1.1.3" xref="alg0.l6.m1.1.1.3.cmml"><mi id="alg0.l6.m1.1.1.3.2" xref="alg0.l6.m1.1.1.3.2.cmml">r</mi><mi id="alg0.l6.m1.1.1.3.3" xref="alg0.l6.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg0.l6.m1.1b"><apply id="alg0.l6.m1.1.1.cmml" xref="alg0.l6.m1.1.1"><csymbol cd="latexml" id="alg0.l6.m1.1.1.1.cmml" xref="alg0.l6.m1.1.1.1">direct-sum</csymbol><apply id="alg0.l6.m1.1.1.2.cmml" xref="alg0.l6.m1.1.1.2"><csymbol cd="ambiguous" id="alg0.l6.m1.1.1.2.1.cmml" xref="alg0.l6.m1.1.1.2">subscript</csymbol><ci id="alg0.l6.m1.1.1.2.2.cmml" xref="alg0.l6.m1.1.1.2.2">𝜏</ci><apply id="alg0.l6.m1.1.1.2.3.cmml" xref="alg0.l6.m1.1.1.2.3"><minus id="alg0.l6.m1.1.1.2.3.1.cmml" xref="alg0.l6.m1.1.1.2.3.1"></minus><ci id="alg0.l6.m1.1.1.2.3.2.cmml" xref="alg0.l6.m1.1.1.2.3.2">𝑖</ci><cn type="integer" id="alg0.l6.m1.1.1.2.3.3.cmml" xref="alg0.l6.m1.1.1.2.3.3">1</cn></apply></apply><apply id="alg0.l6.m1.1.1.3.cmml" xref="alg0.l6.m1.1.1.3"><csymbol cd="ambiguous" id="alg0.l6.m1.1.1.3.1.cmml" xref="alg0.l6.m1.1.1.3">subscript</csymbol><ci id="alg0.l6.m1.1.1.3.2.cmml" xref="alg0.l6.m1.1.1.3.2">𝑟</ci><ci id="alg0.l6.m1.1.1.3.3.cmml" xref="alg0.l6.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg0.l6.m1.1c">\tau_{i-1}\oplus r_{i}</annotation></semantics></math>

</div>
<div id="alg0.l7" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline">7:</span>     <span id="alg0.l7.1" class="ltx_text ltx_font_bold">end</span> <span id="alg0.l7.2" class="ltx_text ltx_font_bold">if</span>
</div>
<div id="alg0.l8" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline">8:</span>     <math id="alg0.l8.m1.1" class="ltx_math_unparsed" alttext="a_{i}\sim\mathbb{P}_{\mathcal{G}}(\cdot|p\oplus q\oplus\tau_{i-1}\oplus r_{i})" display="inline"><semantics id="alg0.l8.m1.1a"><mrow id="alg0.l8.m1.1b"><msub id="alg0.l8.m1.1.1"><mi id="alg0.l8.m1.1.1.2">a</mi><mi id="alg0.l8.m1.1.1.3">i</mi></msub><mo id="alg0.l8.m1.1.2">∼</mo><msub id="alg0.l8.m1.1.3"><mi id="alg0.l8.m1.1.3.2">ℙ</mi><mi class="ltx_font_mathcaligraphic" id="alg0.l8.m1.1.3.3">𝒢</mi></msub><mrow id="alg0.l8.m1.1.4"><mo stretchy="false" id="alg0.l8.m1.1.4.1">(</mo><mo lspace="0em" rspace="0em" id="alg0.l8.m1.1.4.2">⋅</mo><mo fence="false" rspace="0.167em" stretchy="false" id="alg0.l8.m1.1.4.3">|</mo><mi id="alg0.l8.m1.1.4.4">p</mi><mo id="alg0.l8.m1.1.4.5">⊕</mo><mi id="alg0.l8.m1.1.4.6">q</mi><mo id="alg0.l8.m1.1.4.7">⊕</mo><msub id="alg0.l8.m1.1.4.8"><mi id="alg0.l8.m1.1.4.8.2">τ</mi><mrow id="alg0.l8.m1.1.4.8.3"><mi id="alg0.l8.m1.1.4.8.3.2">i</mi><mo id="alg0.l8.m1.1.4.8.3.1">−</mo><mn id="alg0.l8.m1.1.4.8.3.3">1</mn></mrow></msub><mo id="alg0.l8.m1.1.4.9">⊕</mo><msub id="alg0.l8.m1.1.4.10"><mi id="alg0.l8.m1.1.4.10.2">r</mi><mi id="alg0.l8.m1.1.4.10.3">i</mi></msub><mo stretchy="false" id="alg0.l8.m1.1.4.11">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="alg0.l8.m1.1c">a_{i}\sim\mathbb{P}_{\mathcal{G}}(\cdot|p\oplus q\oplus\tau_{i-1}\oplus r_{i})</annotation></semantics></math> <span id="alg0.l8.1" class="ltx_text" style="float:right;"><math id="alg0.l8.1.m1.1" class="ltx_Math" alttext="\triangleright" display="inline"><semantics id="alg0.l8.1.m1.1a"><mo id="alg0.l8.1.m1.1.1" xref="alg0.l8.1.m1.1.1.cmml">▷</mo><annotation-xml encoding="MathML-Content" id="alg0.l8.1.m1.1b"><ci id="alg0.l8.1.m1.1.1.cmml" xref="alg0.l8.1.m1.1.1">▷</ci></annotation-xml><annotation encoding="application/x-tex" id="alg0.l8.1.m1.1c">\triangleright</annotation></semantics></math> Program Generation (Eq. <a href="#S2.E2" title="In Inference Procedure ‣ 2.2 Collecting Interactive Tool-Use Trajectories ‣ 2 ToRA: Tool-Integrated Agents for Mathematical Reasoning ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>)
</span>
</div>
<div id="alg0.l9" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline">9:</span>     <math id="alg0.l9.m1.1" class="ltx_Math" alttext="o_{i}\leftarrow\mathcal{E}(a_{i})" display="inline"><semantics id="alg0.l9.m1.1a"><mrow id="alg0.l9.m1.1.1" xref="alg0.l9.m1.1.1.cmml"><msub id="alg0.l9.m1.1.1.3" xref="alg0.l9.m1.1.1.3.cmml"><mi id="alg0.l9.m1.1.1.3.2" xref="alg0.l9.m1.1.1.3.2.cmml">o</mi><mi id="alg0.l9.m1.1.1.3.3" xref="alg0.l9.m1.1.1.3.3.cmml">i</mi></msub><mo stretchy="false" id="alg0.l9.m1.1.1.2" xref="alg0.l9.m1.1.1.2.cmml">←</mo><mrow id="alg0.l9.m1.1.1.1" xref="alg0.l9.m1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="alg0.l9.m1.1.1.1.3" xref="alg0.l9.m1.1.1.1.3.cmml">ℰ</mi><mo lspace="0em" rspace="0em" id="alg0.l9.m1.1.1.1.2" xref="alg0.l9.m1.1.1.1.2.cmml">​</mo><mrow id="alg0.l9.m1.1.1.1.1.1" xref="alg0.l9.m1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="alg0.l9.m1.1.1.1.1.1.2" xref="alg0.l9.m1.1.1.1.1.1.1.cmml">(</mo><msub id="alg0.l9.m1.1.1.1.1.1.1" xref="alg0.l9.m1.1.1.1.1.1.1.cmml"><mi id="alg0.l9.m1.1.1.1.1.1.1.2" xref="alg0.l9.m1.1.1.1.1.1.1.2.cmml">a</mi><mi id="alg0.l9.m1.1.1.1.1.1.1.3" xref="alg0.l9.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo stretchy="false" id="alg0.l9.m1.1.1.1.1.1.3" xref="alg0.l9.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg0.l9.m1.1b"><apply id="alg0.l9.m1.1.1.cmml" xref="alg0.l9.m1.1.1"><ci id="alg0.l9.m1.1.1.2.cmml" xref="alg0.l9.m1.1.1.2">←</ci><apply id="alg0.l9.m1.1.1.3.cmml" xref="alg0.l9.m1.1.1.3"><csymbol cd="ambiguous" id="alg0.l9.m1.1.1.3.1.cmml" xref="alg0.l9.m1.1.1.3">subscript</csymbol><ci id="alg0.l9.m1.1.1.3.2.cmml" xref="alg0.l9.m1.1.1.3.2">𝑜</ci><ci id="alg0.l9.m1.1.1.3.3.cmml" xref="alg0.l9.m1.1.1.3.3">𝑖</ci></apply><apply id="alg0.l9.m1.1.1.1.cmml" xref="alg0.l9.m1.1.1.1"><times id="alg0.l9.m1.1.1.1.2.cmml" xref="alg0.l9.m1.1.1.1.2"></times><ci id="alg0.l9.m1.1.1.1.3.cmml" xref="alg0.l9.m1.1.1.1.3">ℰ</ci><apply id="alg0.l9.m1.1.1.1.1.1.1.cmml" xref="alg0.l9.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg0.l9.m1.1.1.1.1.1.1.1.cmml" xref="alg0.l9.m1.1.1.1.1.1">subscript</csymbol><ci id="alg0.l9.m1.1.1.1.1.1.1.2.cmml" xref="alg0.l9.m1.1.1.1.1.1.1.2">𝑎</ci><ci id="alg0.l9.m1.1.1.1.1.1.1.3.cmml" xref="alg0.l9.m1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg0.l9.m1.1c">o_{i}\leftarrow\mathcal{E}(a_{i})</annotation></semantics></math> <span id="alg0.l9.1" class="ltx_text" style="float:right;"><math id="alg0.l9.1.m1.1" class="ltx_Math" alttext="\triangleright" display="inline"><semantics id="alg0.l9.1.m1.1a"><mo id="alg0.l9.1.m1.1.1" xref="alg0.l9.1.m1.1.1.cmml">▷</mo><annotation-xml encoding="MathML-Content" id="alg0.l9.1.m1.1b"><ci id="alg0.l9.1.m1.1.1.cmml" xref="alg0.l9.1.m1.1.1">▷</ci></annotation-xml><annotation encoding="application/x-tex" id="alg0.l9.1.m1.1c">\triangleright</annotation></semantics></math> Tool Execution
</span>
</div>
<div id="alg0.l10" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline">10:</span>     <math id="alg0.l10.m1.1" class="ltx_Math" alttext="\tau_{i}\leftarrow\tau_{i-1}\oplus r_{i}\oplus a_{i}\oplus o_{i}" display="inline"><semantics id="alg0.l10.m1.1a"><mrow id="alg0.l10.m1.1.1" xref="alg0.l10.m1.1.1.cmml"><msub id="alg0.l10.m1.1.1.2" xref="alg0.l10.m1.1.1.2.cmml"><mi id="alg0.l10.m1.1.1.2.2" xref="alg0.l10.m1.1.1.2.2.cmml">τ</mi><mi id="alg0.l10.m1.1.1.2.3" xref="alg0.l10.m1.1.1.2.3.cmml">i</mi></msub><mo stretchy="false" id="alg0.l10.m1.1.1.1" xref="alg0.l10.m1.1.1.1.cmml">←</mo><mrow id="alg0.l10.m1.1.1.3" xref="alg0.l10.m1.1.1.3.cmml"><msub id="alg0.l10.m1.1.1.3.2" xref="alg0.l10.m1.1.1.3.2.cmml"><mi id="alg0.l10.m1.1.1.3.2.2" xref="alg0.l10.m1.1.1.3.2.2.cmml">τ</mi><mrow id="alg0.l10.m1.1.1.3.2.3" xref="alg0.l10.m1.1.1.3.2.3.cmml"><mi id="alg0.l10.m1.1.1.3.2.3.2" xref="alg0.l10.m1.1.1.3.2.3.2.cmml">i</mi><mo id="alg0.l10.m1.1.1.3.2.3.1" xref="alg0.l10.m1.1.1.3.2.3.1.cmml">−</mo><mn id="alg0.l10.m1.1.1.3.2.3.3" xref="alg0.l10.m1.1.1.3.2.3.3.cmml">1</mn></mrow></msub><mo id="alg0.l10.m1.1.1.3.1" xref="alg0.l10.m1.1.1.3.1.cmml">⊕</mo><msub id="alg0.l10.m1.1.1.3.3" xref="alg0.l10.m1.1.1.3.3.cmml"><mi id="alg0.l10.m1.1.1.3.3.2" xref="alg0.l10.m1.1.1.3.3.2.cmml">r</mi><mi id="alg0.l10.m1.1.1.3.3.3" xref="alg0.l10.m1.1.1.3.3.3.cmml">i</mi></msub><mo id="alg0.l10.m1.1.1.3.1a" xref="alg0.l10.m1.1.1.3.1.cmml">⊕</mo><msub id="alg0.l10.m1.1.1.3.4" xref="alg0.l10.m1.1.1.3.4.cmml"><mi id="alg0.l10.m1.1.1.3.4.2" xref="alg0.l10.m1.1.1.3.4.2.cmml">a</mi><mi id="alg0.l10.m1.1.1.3.4.3" xref="alg0.l10.m1.1.1.3.4.3.cmml">i</mi></msub><mo id="alg0.l10.m1.1.1.3.1b" xref="alg0.l10.m1.1.1.3.1.cmml">⊕</mo><msub id="alg0.l10.m1.1.1.3.5" xref="alg0.l10.m1.1.1.3.5.cmml"><mi id="alg0.l10.m1.1.1.3.5.2" xref="alg0.l10.m1.1.1.3.5.2.cmml">o</mi><mi id="alg0.l10.m1.1.1.3.5.3" xref="alg0.l10.m1.1.1.3.5.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg0.l10.m1.1b"><apply id="alg0.l10.m1.1.1.cmml" xref="alg0.l10.m1.1.1"><ci id="alg0.l10.m1.1.1.1.cmml" xref="alg0.l10.m1.1.1.1">←</ci><apply id="alg0.l10.m1.1.1.2.cmml" xref="alg0.l10.m1.1.1.2"><csymbol cd="ambiguous" id="alg0.l10.m1.1.1.2.1.cmml" xref="alg0.l10.m1.1.1.2">subscript</csymbol><ci id="alg0.l10.m1.1.1.2.2.cmml" xref="alg0.l10.m1.1.1.2.2">𝜏</ci><ci id="alg0.l10.m1.1.1.2.3.cmml" xref="alg0.l10.m1.1.1.2.3">𝑖</ci></apply><apply id="alg0.l10.m1.1.1.3.cmml" xref="alg0.l10.m1.1.1.3"><csymbol cd="latexml" id="alg0.l10.m1.1.1.3.1.cmml" xref="alg0.l10.m1.1.1.3.1">direct-sum</csymbol><apply id="alg0.l10.m1.1.1.3.2.cmml" xref="alg0.l10.m1.1.1.3.2"><csymbol cd="ambiguous" id="alg0.l10.m1.1.1.3.2.1.cmml" xref="alg0.l10.m1.1.1.3.2">subscript</csymbol><ci id="alg0.l10.m1.1.1.3.2.2.cmml" xref="alg0.l10.m1.1.1.3.2.2">𝜏</ci><apply id="alg0.l10.m1.1.1.3.2.3.cmml" xref="alg0.l10.m1.1.1.3.2.3"><minus id="alg0.l10.m1.1.1.3.2.3.1.cmml" xref="alg0.l10.m1.1.1.3.2.3.1"></minus><ci id="alg0.l10.m1.1.1.3.2.3.2.cmml" xref="alg0.l10.m1.1.1.3.2.3.2">𝑖</ci><cn type="integer" id="alg0.l10.m1.1.1.3.2.3.3.cmml" xref="alg0.l10.m1.1.1.3.2.3.3">1</cn></apply></apply><apply id="alg0.l10.m1.1.1.3.3.cmml" xref="alg0.l10.m1.1.1.3.3"><csymbol cd="ambiguous" id="alg0.l10.m1.1.1.3.3.1.cmml" xref="alg0.l10.m1.1.1.3.3">subscript</csymbol><ci id="alg0.l10.m1.1.1.3.3.2.cmml" xref="alg0.l10.m1.1.1.3.3.2">𝑟</ci><ci id="alg0.l10.m1.1.1.3.3.3.cmml" xref="alg0.l10.m1.1.1.3.3.3">𝑖</ci></apply><apply id="alg0.l10.m1.1.1.3.4.cmml" xref="alg0.l10.m1.1.1.3.4"><csymbol cd="ambiguous" id="alg0.l10.m1.1.1.3.4.1.cmml" xref="alg0.l10.m1.1.1.3.4">subscript</csymbol><ci id="alg0.l10.m1.1.1.3.4.2.cmml" xref="alg0.l10.m1.1.1.3.4.2">𝑎</ci><ci id="alg0.l10.m1.1.1.3.4.3.cmml" xref="alg0.l10.m1.1.1.3.4.3">𝑖</ci></apply><apply id="alg0.l10.m1.1.1.3.5.cmml" xref="alg0.l10.m1.1.1.3.5"><csymbol cd="ambiguous" id="alg0.l10.m1.1.1.3.5.1.cmml" xref="alg0.l10.m1.1.1.3.5">subscript</csymbol><ci id="alg0.l10.m1.1.1.3.5.2.cmml" xref="alg0.l10.m1.1.1.3.5.2">𝑜</ci><ci id="alg0.l10.m1.1.1.3.5.3.cmml" xref="alg0.l10.m1.1.1.3.5.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg0.l10.m1.1c">\tau_{i}\leftarrow\tau_{i-1}\oplus r_{i}\oplus a_{i}\oplus o_{i}</annotation></semantics></math> <span id="alg0.l10.1" class="ltx_text" style="float:right;"><math id="alg0.l10.1.m1.1" class="ltx_Math" alttext="\triangleright" display="inline"><semantics id="alg0.l10.1.m1.1a"><mo id="alg0.l10.1.m1.1.1" xref="alg0.l10.1.m1.1.1.cmml">▷</mo><annotation-xml encoding="MathML-Content" id="alg0.l10.1.m1.1b"><ci id="alg0.l10.1.m1.1.1.cmml" xref="alg0.l10.1.m1.1.1">▷</ci></annotation-xml><annotation encoding="application/x-tex" id="alg0.l10.1.m1.1c">\triangleright</annotation></semantics></math> Trajectory Update (Eq. <a href="#S2.E3" title="In Inference Procedure ‣ 2.2 Collecting Interactive Tool-Use Trajectories ‣ 2 ToRA: Tool-Integrated Agents for Mathematical Reasoning ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3</span></a>)
</span>
</div>
<div id="alg0.l11" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline">11:</span><span id="alg0.l11.1" class="ltx_text ltx_font_bold">end</span> <span id="alg0.l11.2" class="ltx_text ltx_font_bold">for</span>
</div>
<div id="alg0.l12" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline">12:</span><span id="alg0.l12.1" class="ltx_text ltx_font_bold">return</span> <math id="alg0.l12.m1.1" class="ltx_Math" alttext="\tau_{n}" display="inline"><semantics id="alg0.l12.m1.1a"><msub id="alg0.l12.m1.1.1" xref="alg0.l12.m1.1.1.cmml"><mi id="alg0.l12.m1.1.1.2" xref="alg0.l12.m1.1.1.2.cmml">τ</mi><mi id="alg0.l12.m1.1.1.3" xref="alg0.l12.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="alg0.l12.m1.1b"><apply id="alg0.l12.m1.1.1.cmml" xref="alg0.l12.m1.1.1"><csymbol cd="ambiguous" id="alg0.l12.m1.1.1.1.cmml" xref="alg0.l12.m1.1.1">subscript</csymbol><ci id="alg0.l12.m1.1.1.2.cmml" xref="alg0.l12.m1.1.1.2">𝜏</ci><ci id="alg0.l12.m1.1.1.3.cmml" xref="alg0.l12.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg0.l12.m1.1c">\tau_{n}</annotation></semantics></math>

</div>
</div>
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span id="alg1.3.1.1" class="ltx_text ltx_font_bold">Algorithm 1</span> </span> Inference of Tool-Integrated Reasoning</figcaption>
</figure>
</figure>
<div id="S2.SS2.p1" class="ltx_para ltx_noindent">
<p id="S2.SS2.p1.1" class="ltx_p">Existing mathematical reasoning datasets primarily contain annotations in either natural language or code, posing a challenge for training tool-integrated agents due to the absence of interactive tool-use annotations.
To address this, we utilize GPT-4 to synthesize high-quality trajectories on the GSM8k and MATH training sets. We select GSM8k and MATH as they exhibit diverse reasoning patterns, spanning multiple domains and difficulty levels.</p>
</div>
<section id="S2.SS2.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Prompt Curation</h5>

<div id="S2.SS2.SSS0.Px1.p1" class="ltx_para ltx_noindent">
<p id="S2.SS2.SSS0.Px1.p1.1" class="ltx_p">We compose instructions along with diverse few-shot examples, utilizing an interleaved format as depicted in Fig <a href="#S0.F2" title="Figure 2 ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a> (c). These examples showcase interactive tool usage trajectories, incorporating descriptive variable names and combined program outputs. Please refer to Appendix <a href="#A5" title="Appendix E Prompts ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">E</span></a> for the assembled prompts.</p>
</div>
</section>
<section id="S2.SS2.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Inference Procedure</h5>

<div id="S2.SS2.SSS0.Px2.p1" class="ltx_para ltx_noindent">
<p id="S2.SS2.SSS0.Px2.p1.6" class="ltx_p">We follow Algorithm <a href="#alg1" title="Algorithm 1 ‣ 2.2 Collecting Interactive Tool-Use Trajectories ‣ 2 ToRA: Tool-Integrated Agents for Mathematical Reasoning ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a> and feed GPT-4 (<math id="S2.SS2.SSS0.Px2.p1.1.m1.1" class="ltx_Math" alttext="\mathcal{G}" display="inline"><semantics id="S2.SS2.SSS0.Px2.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS0.Px2.p1.1.m1.1.1" xref="S2.SS2.SSS0.Px2.p1.1.m1.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.1.m1.1b"><ci id="S2.SS2.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.1.m1.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.1.m1.1c">\mathcal{G}</annotation></semantics></math>) with the composed prompt <math id="S2.SS2.SSS0.Px2.p1.2.m2.1" class="ltx_Math" alttext="p" display="inline"><semantics id="S2.SS2.SSS0.Px2.p1.2.m2.1a"><mi id="S2.SS2.SSS0.Px2.p1.2.m2.1.1" xref="S2.SS2.SSS0.Px2.p1.2.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.2.m2.1b"><ci id="S2.SS2.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.2.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.2.m2.1c">p</annotation></semantics></math> to generate a tool-use trajectory <math id="S2.SS2.SSS0.Px2.p1.3.m3.1" class="ltx_Math" alttext="\tau" display="inline"><semantics id="S2.SS2.SSS0.Px2.p1.3.m3.1a"><mi id="S2.SS2.SSS0.Px2.p1.3.m3.1.1" xref="S2.SS2.SSS0.Px2.p1.3.m3.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.3.m3.1b"><ci id="S2.SS2.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.3.m3.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.3.m3.1c">\tau</annotation></semantics></math> for each question <math id="S2.SS2.SSS0.Px2.p1.4.m4.1" class="ltx_Math" alttext="q" display="inline"><semantics id="S2.SS2.SSS0.Px2.p1.4.m4.1a"><mi id="S2.SS2.SSS0.Px2.p1.4.m4.1.1" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.4.m4.1b"><ci id="S2.SS2.SSS0.Px2.p1.4.m4.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.4.m4.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.4.m4.1c">q</annotation></semantics></math> from the training set.
The trajectory is initialized as an empty string <math id="S2.SS2.SSS0.Px2.p1.5.m5.1" class="ltx_Math" alttext="\tau_{0}" display="inline"><semantics id="S2.SS2.SSS0.Px2.p1.5.m5.1a"><msub id="S2.SS2.SSS0.Px2.p1.5.m5.1.1" xref="S2.SS2.SSS0.Px2.p1.5.m5.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p1.5.m5.1.1.2" xref="S2.SS2.SSS0.Px2.p1.5.m5.1.1.2.cmml">τ</mi><mn id="S2.SS2.SSS0.Px2.p1.5.m5.1.1.3" xref="S2.SS2.SSS0.Px2.p1.5.m5.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.5.m5.1b"><apply id="S2.SS2.SSS0.Px2.p1.5.m5.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.5.m5.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.5.m5.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.5.m5.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.5.m5.1.1.2">𝜏</ci><cn type="integer" id="S2.SS2.SSS0.Px2.p1.5.m5.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p1.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.5.m5.1c">\tau_{0}</annotation></semantics></math>, for each interaction round <math id="S2.SS2.SSS0.Px2.p1.6.m6.1" class="ltx_Math" alttext="i" display="inline"><semantics id="S2.SS2.SSS0.Px2.p1.6.m6.1a"><mi id="S2.SS2.SSS0.Px2.p1.6.m6.1.1" xref="S2.SS2.SSS0.Px2.p1.6.m6.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.6.m6.1b"><ci id="S2.SS2.SSS0.Px2.p1.6.m6.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.6.m6.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.6.m6.1c">i</annotation></semantics></math>, we first generate a rationale:</p>
<table id="A6.EGx1" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="S2.E1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="S2.E1.m1.1" class="ltx_math_unparsed" alttext="\displaystyle r_{i}\sim\mathbb{P}_{\mathcal{G}}(\cdot|p\oplus q\oplus\tau_{i-1})" display="inline"><semantics id="S2.E1.m1.1a"><mrow id="S2.E1.m1.1b"><msub id="S2.E1.m1.1.1"><mi id="S2.E1.m1.1.1.2">r</mi><mi id="S2.E1.m1.1.1.3">i</mi></msub><mo id="S2.E1.m1.1.2">∼</mo><msub id="S2.E1.m1.1.3"><mi id="S2.E1.m1.1.3.2">ℙ</mi><mi class="ltx_font_mathcaligraphic" id="S2.E1.m1.1.3.3">𝒢</mi></msub><mrow id="S2.E1.m1.1.4"><mo stretchy="false" id="S2.E1.m1.1.4.1">(</mo><mo lspace="0em" rspace="0em" id="S2.E1.m1.1.4.2">⋅</mo><mo fence="false" rspace="0.167em" stretchy="false" id="S2.E1.m1.1.4.3">|</mo><mi id="S2.E1.m1.1.4.4">p</mi><mo id="S2.E1.m1.1.4.5">⊕</mo><mi id="S2.E1.m1.1.4.6">q</mi><mo id="S2.E1.m1.1.4.7">⊕</mo><msub id="S2.E1.m1.1.4.8"><mi id="S2.E1.m1.1.4.8.2">τ</mi><mrow id="S2.E1.m1.1.4.8.3"><mi id="S2.E1.m1.1.4.8.3.2">i</mi><mo id="S2.E1.m1.1.4.8.3.1">−</mo><mn id="S2.E1.m1.1.4.8.3.3">1</mn></mrow></msub><mo stretchy="false" id="S2.E1.m1.1.4.9">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.E1.m1.1c">\displaystyle r_{i}\sim\mathbb{P}_{\mathcal{G}}(\cdot|p\oplus q\oplus\tau_{i-1})</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td>
</tr></tbody>
</table>
<p id="S2.SS2.SSS0.Px2.p1.9" class="ltx_p">where <math id="S2.SS2.SSS0.Px2.p1.7.m1.1" class="ltx_Math" alttext="\oplus" display="inline"><semantics id="S2.SS2.SSS0.Px2.p1.7.m1.1a"><mo id="S2.SS2.SSS0.Px2.p1.7.m1.1.1" xref="S2.SS2.SSS0.Px2.p1.7.m1.1.1.cmml">⊕</mo><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.7.m1.1b"><csymbol cd="latexml" id="S2.SS2.SSS0.Px2.p1.7.m1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.7.m1.1.1">direct-sum</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.7.m1.1c">\oplus</annotation></semantics></math> means concatenation. If <math id="S2.SS2.SSS0.Px2.p1.8.m2.1" class="ltx_Math" alttext="r_{i}" display="inline"><semantics id="S2.SS2.SSS0.Px2.p1.8.m2.1a"><msub id="S2.SS2.SSS0.Px2.p1.8.m2.1.1" xref="S2.SS2.SSS0.Px2.p1.8.m2.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p1.8.m2.1.1.2" xref="S2.SS2.SSS0.Px2.p1.8.m2.1.1.2.cmml">r</mi><mi id="S2.SS2.SSS0.Px2.p1.8.m2.1.1.3" xref="S2.SS2.SSS0.Px2.p1.8.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.8.m2.1b"><apply id="S2.SS2.SSS0.Px2.p1.8.m2.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.8.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.8.m2.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.8.m2.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.8.m2.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.8.m2.1.1.2">𝑟</ci><ci id="S2.SS2.SSS0.Px2.p1.8.m2.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p1.8.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.8.m2.1c">r_{i}</annotation></semantics></math> includes an answer within “<span id="S2.SS2.SSS0.Px2.p1.9.1" class="ltx_text ltx_lstlisting ltx_font_typewriter ltx_framed ltx_framed_topbottom">\<span id="S2.SS2.SSS0.Px2.p1.9.1.1" class="ltx_text ltx_lst_identifier">boxed</span>{}</span>” (i.e., the stopping condition <span id="S2.SS2.SSS0.Px2.p1.9.2" class="ltx_text ltx_font_italic">Stop</span>(<math id="S2.SS2.SSS0.Px2.p1.9.m3.1" class="ltx_Math" alttext="r_{i}" display="inline"><semantics id="S2.SS2.SSS0.Px2.p1.9.m3.1a"><msub id="S2.SS2.SSS0.Px2.p1.9.m3.1.1" xref="S2.SS2.SSS0.Px2.p1.9.m3.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p1.9.m3.1.1.2" xref="S2.SS2.SSS0.Px2.p1.9.m3.1.1.2.cmml">r</mi><mi id="S2.SS2.SSS0.Px2.p1.9.m3.1.1.3" xref="S2.SS2.SSS0.Px2.p1.9.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.9.m3.1b"><apply id="S2.SS2.SSS0.Px2.p1.9.m3.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.9.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.9.m3.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.9.m3.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.9.m3.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.9.m3.1.1.2">𝑟</ci><ci id="S2.SS2.SSS0.Px2.p1.9.m3.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p1.9.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.9.m3.1c">r_{i}</annotation></semantics></math>)), we cease generation, otherwise the model continues to write a program for tool use:</p>
<table id="A6.EGx2" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="S2.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="S2.E2.m1.1" class="ltx_math_unparsed" alttext="\displaystyle a_{i}\sim\mathbb{P}_{\mathcal{G}}(\cdot|p\oplus q\oplus\tau_{i-1}\oplus r_{i})" display="inline"><semantics id="S2.E2.m1.1a"><mrow id="S2.E2.m1.1b"><msub id="S2.E2.m1.1.1"><mi id="S2.E2.m1.1.1.2">a</mi><mi id="S2.E2.m1.1.1.3">i</mi></msub><mo id="S2.E2.m1.1.2">∼</mo><msub id="S2.E2.m1.1.3"><mi id="S2.E2.m1.1.3.2">ℙ</mi><mi class="ltx_font_mathcaligraphic" id="S2.E2.m1.1.3.3">𝒢</mi></msub><mrow id="S2.E2.m1.1.4"><mo stretchy="false" id="S2.E2.m1.1.4.1">(</mo><mo lspace="0em" rspace="0em" id="S2.E2.m1.1.4.2">⋅</mo><mo fence="false" rspace="0.167em" stretchy="false" id="S2.E2.m1.1.4.3">|</mo><mi id="S2.E2.m1.1.4.4">p</mi><mo id="S2.E2.m1.1.4.5">⊕</mo><mi id="S2.E2.m1.1.4.6">q</mi><mo id="S2.E2.m1.1.4.7">⊕</mo><msub id="S2.E2.m1.1.4.8"><mi id="S2.E2.m1.1.4.8.2">τ</mi><mrow id="S2.E2.m1.1.4.8.3"><mi id="S2.E2.m1.1.4.8.3.2">i</mi><mo id="S2.E2.m1.1.4.8.3.1">−</mo><mn id="S2.E2.m1.1.4.8.3.3">1</mn></mrow></msub><mo id="S2.E2.m1.1.4.9">⊕</mo><msub id="S2.E2.m1.1.4.10"><mi id="S2.E2.m1.1.4.10.2">r</mi><mi id="S2.E2.m1.1.4.10.3">i</mi></msub><mo stretchy="false" id="S2.E2.m1.1.4.11">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.E2.m1.1c">\displaystyle a_{i}\sim\mathbb{P}_{\mathcal{G}}(\cdot|p\oplus q\oplus\tau_{i-1}\oplus r_{i})</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td>
</tr></tbody>
</table>
<p id="S2.SS2.SSS0.Px2.p1.14" class="ltx_p">In line with <cite class="ltx_cite ltx_citemacro_citet">Gou et al. (<a href="#bib.bib11" title="" class="ltx_ref">2023</a>)</cite>, if the model triggers the code execution stop words like “<span id="S2.SS2.SSS0.Px2.p1.14.1" class="ltx_text ltx_lstlisting ltx_font_typewriter ltx_framed ltx_framed_topbottom">‘‘‘<span id="S2.SS2.SSS0.Px2.p1.14.1.1" class="ltx_text ltx_lst_identifier">output</span></span>”, we supply it with the corresponding execution message and output <math id="S2.SS2.SSS0.Px2.p1.10.m1.1" class="ltx_Math" alttext="o_{i}" display="inline"><semantics id="S2.SS2.SSS0.Px2.p1.10.m1.1a"><msub id="S2.SS2.SSS0.Px2.p1.10.m1.1.1" xref="S2.SS2.SSS0.Px2.p1.10.m1.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p1.10.m1.1.1.2" xref="S2.SS2.SSS0.Px2.p1.10.m1.1.1.2.cmml">o</mi><mi id="S2.SS2.SSS0.Px2.p1.10.m1.1.1.3" xref="S2.SS2.SSS0.Px2.p1.10.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.10.m1.1b"><apply id="S2.SS2.SSS0.Px2.p1.10.m1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.10.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.10.m1.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.10.m1.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.10.m1.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.10.m1.1.1.2">𝑜</ci><ci id="S2.SS2.SSS0.Px2.p1.10.m1.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p1.10.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.10.m1.1c">o_{i}</annotation></semantics></math> by calling tools with <math id="S2.SS2.SSS0.Px2.p1.11.m2.1" class="ltx_Math" alttext="o_{i}\leftarrow\mathcal{E}(a_{i})" display="inline"><semantics id="S2.SS2.SSS0.Px2.p1.11.m2.1a"><mrow id="S2.SS2.SSS0.Px2.p1.11.m2.1.1" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.cmml"><msub id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.3" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.3.2" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.3.2.cmml">o</mi><mi id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.3.3" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.3.3.cmml">i</mi></msub><mo stretchy="false" id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.2" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.2.cmml">←</mo><mrow id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.3" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.3.cmml">ℰ</mi><mo lspace="0em" rspace="0em" id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.2" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.2.cmml">​</mo><mrow id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1.2" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1.1" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1.1.2" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1.1.2.cmml">a</mi><mi id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1.1.3" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1.1.3.cmml">i</mi></msub><mo stretchy="false" id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1.3" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.11.m2.1b"><apply id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1"><ci id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.2">←</ci><apply id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.3">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.3.2">𝑜</ci><ci id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.3.3.cmml" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.3.3">𝑖</ci></apply><apply id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1"><times id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.2"></times><ci id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.3">ℰ</ci><apply id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1.1.2">𝑎</ci><ci id="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p1.11.m2.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.11.m2.1c">o_{i}\leftarrow\mathcal{E}(a_{i})</annotation></semantics></math>, facilitating the generation of subsequent steps. Then, we update the trajectory by concatenating it with the newly generated rationale <math id="S2.SS2.SSS0.Px2.p1.12.m3.1" class="ltx_Math" alttext="r_{i}" display="inline"><semantics id="S2.SS2.SSS0.Px2.p1.12.m3.1a"><msub id="S2.SS2.SSS0.Px2.p1.12.m3.1.1" xref="S2.SS2.SSS0.Px2.p1.12.m3.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p1.12.m3.1.1.2" xref="S2.SS2.SSS0.Px2.p1.12.m3.1.1.2.cmml">r</mi><mi id="S2.SS2.SSS0.Px2.p1.12.m3.1.1.3" xref="S2.SS2.SSS0.Px2.p1.12.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.12.m3.1b"><apply id="S2.SS2.SSS0.Px2.p1.12.m3.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.12.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.12.m3.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.12.m3.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.12.m3.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.12.m3.1.1.2">𝑟</ci><ci id="S2.SS2.SSS0.Px2.p1.12.m3.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p1.12.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.12.m3.1c">r_{i}</annotation></semantics></math>, program <math id="S2.SS2.SSS0.Px2.p1.13.m4.1" class="ltx_Math" alttext="a_{i}" display="inline"><semantics id="S2.SS2.SSS0.Px2.p1.13.m4.1a"><msub id="S2.SS2.SSS0.Px2.p1.13.m4.1.1" xref="S2.SS2.SSS0.Px2.p1.13.m4.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p1.13.m4.1.1.2" xref="S2.SS2.SSS0.Px2.p1.13.m4.1.1.2.cmml">a</mi><mi id="S2.SS2.SSS0.Px2.p1.13.m4.1.1.3" xref="S2.SS2.SSS0.Px2.p1.13.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.13.m4.1b"><apply id="S2.SS2.SSS0.Px2.p1.13.m4.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.13.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.13.m4.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.13.m4.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.13.m4.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.13.m4.1.1.2">𝑎</ci><ci id="S2.SS2.SSS0.Px2.p1.13.m4.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p1.13.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.13.m4.1c">a_{i}</annotation></semantics></math>, and output <math id="S2.SS2.SSS0.Px2.p1.14.m5.1" class="ltx_Math" alttext="o_{i}" display="inline"><semantics id="S2.SS2.SSS0.Px2.p1.14.m5.1a"><msub id="S2.SS2.SSS0.Px2.p1.14.m5.1.1" xref="S2.SS2.SSS0.Px2.p1.14.m5.1.1.cmml"><mi id="S2.SS2.SSS0.Px2.p1.14.m5.1.1.2" xref="S2.SS2.SSS0.Px2.p1.14.m5.1.1.2.cmml">o</mi><mi id="S2.SS2.SSS0.Px2.p1.14.m5.1.1.3" xref="S2.SS2.SSS0.Px2.p1.14.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.14.m5.1b"><apply id="S2.SS2.SSS0.Px2.p1.14.m5.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.14.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p1.14.m5.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.14.m5.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p1.14.m5.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p1.14.m5.1.1.2">𝑜</ci><ci id="S2.SS2.SSS0.Px2.p1.14.m5.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p1.14.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.14.m5.1c">o_{i}</annotation></semantics></math>:</p>
<table id="A6.EGx3" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="S2.E3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="S2.E3.m1.1" class="ltx_Math" alttext="\displaystyle\tau_{i}\leftarrow\tau_{i-1}\oplus r_{i}\oplus a_{i}\oplus o_{i}" display="inline"><semantics id="S2.E3.m1.1a"><mrow id="S2.E3.m1.1.1" xref="S2.E3.m1.1.1.cmml"><msub id="S2.E3.m1.1.1.2" xref="S2.E3.m1.1.1.2.cmml"><mi id="S2.E3.m1.1.1.2.2" xref="S2.E3.m1.1.1.2.2.cmml">τ</mi><mi id="S2.E3.m1.1.1.2.3" xref="S2.E3.m1.1.1.2.3.cmml">i</mi></msub><mo stretchy="false" id="S2.E3.m1.1.1.1" xref="S2.E3.m1.1.1.1.cmml">←</mo><mrow id="S2.E3.m1.1.1.3" xref="S2.E3.m1.1.1.3.cmml"><msub id="S2.E3.m1.1.1.3.2" xref="S2.E3.m1.1.1.3.2.cmml"><mi id="S2.E3.m1.1.1.3.2.2" xref="S2.E3.m1.1.1.3.2.2.cmml">τ</mi><mrow id="S2.E3.m1.1.1.3.2.3" xref="S2.E3.m1.1.1.3.2.3.cmml"><mi id="S2.E3.m1.1.1.3.2.3.2" xref="S2.E3.m1.1.1.3.2.3.2.cmml">i</mi><mo id="S2.E3.m1.1.1.3.2.3.1" xref="S2.E3.m1.1.1.3.2.3.1.cmml">−</mo><mn id="S2.E3.m1.1.1.3.2.3.3" xref="S2.E3.m1.1.1.3.2.3.3.cmml">1</mn></mrow></msub><mo id="S2.E3.m1.1.1.3.1" xref="S2.E3.m1.1.1.3.1.cmml">⊕</mo><msub id="S2.E3.m1.1.1.3.3" xref="S2.E3.m1.1.1.3.3.cmml"><mi id="S2.E3.m1.1.1.3.3.2" xref="S2.E3.m1.1.1.3.3.2.cmml">r</mi><mi id="S2.E3.m1.1.1.3.3.3" xref="S2.E3.m1.1.1.3.3.3.cmml">i</mi></msub><mo id="S2.E3.m1.1.1.3.1a" xref="S2.E3.m1.1.1.3.1.cmml">⊕</mo><msub id="S2.E3.m1.1.1.3.4" xref="S2.E3.m1.1.1.3.4.cmml"><mi id="S2.E3.m1.1.1.3.4.2" xref="S2.E3.m1.1.1.3.4.2.cmml">a</mi><mi id="S2.E3.m1.1.1.3.4.3" xref="S2.E3.m1.1.1.3.4.3.cmml">i</mi></msub><mo id="S2.E3.m1.1.1.3.1b" xref="S2.E3.m1.1.1.3.1.cmml">⊕</mo><msub id="S2.E3.m1.1.1.3.5" xref="S2.E3.m1.1.1.3.5.cmml"><mi id="S2.E3.m1.1.1.3.5.2" xref="S2.E3.m1.1.1.3.5.2.cmml">o</mi><mi id="S2.E3.m1.1.1.3.5.3" xref="S2.E3.m1.1.1.3.5.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E3.m1.1b"><apply id="S2.E3.m1.1.1.cmml" xref="S2.E3.m1.1.1"><ci id="S2.E3.m1.1.1.1.cmml" xref="S2.E3.m1.1.1.1">←</ci><apply id="S2.E3.m1.1.1.2.cmml" xref="S2.E3.m1.1.1.2"><csymbol cd="ambiguous" id="S2.E3.m1.1.1.2.1.cmml" xref="S2.E3.m1.1.1.2">subscript</csymbol><ci id="S2.E3.m1.1.1.2.2.cmml" xref="S2.E3.m1.1.1.2.2">𝜏</ci><ci id="S2.E3.m1.1.1.2.3.cmml" xref="S2.E3.m1.1.1.2.3">𝑖</ci></apply><apply id="S2.E3.m1.1.1.3.cmml" xref="S2.E3.m1.1.1.3"><csymbol cd="latexml" id="S2.E3.m1.1.1.3.1.cmml" xref="S2.E3.m1.1.1.3.1">direct-sum</csymbol><apply id="S2.E3.m1.1.1.3.2.cmml" xref="S2.E3.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.E3.m1.1.1.3.2.1.cmml" xref="S2.E3.m1.1.1.3.2">subscript</csymbol><ci id="S2.E3.m1.1.1.3.2.2.cmml" xref="S2.E3.m1.1.1.3.2.2">𝜏</ci><apply id="S2.E3.m1.1.1.3.2.3.cmml" xref="S2.E3.m1.1.1.3.2.3"><minus id="S2.E3.m1.1.1.3.2.3.1.cmml" xref="S2.E3.m1.1.1.3.2.3.1"></minus><ci id="S2.E3.m1.1.1.3.2.3.2.cmml" xref="S2.E3.m1.1.1.3.2.3.2">𝑖</ci><cn type="integer" id="S2.E3.m1.1.1.3.2.3.3.cmml" xref="S2.E3.m1.1.1.3.2.3.3">1</cn></apply></apply><apply id="S2.E3.m1.1.1.3.3.cmml" xref="S2.E3.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.E3.m1.1.1.3.3.1.cmml" xref="S2.E3.m1.1.1.3.3">subscript</csymbol><ci id="S2.E3.m1.1.1.3.3.2.cmml" xref="S2.E3.m1.1.1.3.3.2">𝑟</ci><ci id="S2.E3.m1.1.1.3.3.3.cmml" xref="S2.E3.m1.1.1.3.3.3">𝑖</ci></apply><apply id="S2.E3.m1.1.1.3.4.cmml" xref="S2.E3.m1.1.1.3.4"><csymbol cd="ambiguous" id="S2.E3.m1.1.1.3.4.1.cmml" xref="S2.E3.m1.1.1.3.4">subscript</csymbol><ci id="S2.E3.m1.1.1.3.4.2.cmml" xref="S2.E3.m1.1.1.3.4.2">𝑎</ci><ci id="S2.E3.m1.1.1.3.4.3.cmml" xref="S2.E3.m1.1.1.3.4.3">𝑖</ci></apply><apply id="S2.E3.m1.1.1.3.5.cmml" xref="S2.E3.m1.1.1.3.5"><csymbol cd="ambiguous" id="S2.E3.m1.1.1.3.5.1.cmml" xref="S2.E3.m1.1.1.3.5">subscript</csymbol><ci id="S2.E3.m1.1.1.3.5.2.cmml" xref="S2.E3.m1.1.1.3.5.2">𝑜</ci><ci id="S2.E3.m1.1.1.3.5.3.cmml" xref="S2.E3.m1.1.1.3.5.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.1c">\displaystyle\tau_{i}\leftarrow\tau_{i-1}\oplus r_{i}\oplus a_{i}\oplus o_{i}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td>
</tr></tbody>
</table>
<p id="S2.SS2.SSS0.Px2.p1.15" class="ltx_p">We repeat the above interaction process until we reach the maximum rounds <math id="S2.SS2.SSS0.Px2.p1.15.m1.1" class="ltx_Math" alttext="n" display="inline"><semantics id="S2.SS2.SSS0.Px2.p1.15.m1.1a"><mi id="S2.SS2.SSS0.Px2.p1.15.m1.1.1" xref="S2.SS2.SSS0.Px2.p1.15.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p1.15.m1.1b"><ci id="S2.SS2.SSS0.Px2.p1.15.m1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p1.15.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p1.15.m1.1c">n</annotation></semantics></math>.</p>
</div>
</section>
<section id="S2.SS2.SSS0.Px3" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Trajectory Sampling</h5>

<div id="S2.SS2.SSS0.Px3.p1" class="ltx_para ltx_noindent">
<p id="S2.SS2.SSS0.Px3.p1.1" class="ltx_p">We set <math id="S2.SS2.SSS0.Px3.p1.1.m1.1" class="ltx_Math" alttext="n=3" display="inline"><semantics id="S2.SS2.SSS0.Px3.p1.1.m1.1a"><mrow id="S2.SS2.SSS0.Px3.p1.1.m1.1.1" xref="S2.SS2.SSS0.Px3.p1.1.m1.1.1.cmml"><mi id="S2.SS2.SSS0.Px3.p1.1.m1.1.1.2" xref="S2.SS2.SSS0.Px3.p1.1.m1.1.1.2.cmml">n</mi><mo id="S2.SS2.SSS0.Px3.p1.1.m1.1.1.1" xref="S2.SS2.SSS0.Px3.p1.1.m1.1.1.1.cmml">=</mo><mn id="S2.SS2.SSS0.Px3.p1.1.m1.1.1.3" xref="S2.SS2.SSS0.Px3.p1.1.m1.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px3.p1.1.m1.1b"><apply id="S2.SS2.SSS0.Px3.p1.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.1.m1.1.1"><eq id="S2.SS2.SSS0.Px3.p1.1.m1.1.1.1.cmml" xref="S2.SS2.SSS0.Px3.p1.1.m1.1.1.1"></eq><ci id="S2.SS2.SSS0.Px3.p1.1.m1.1.1.2.cmml" xref="S2.SS2.SSS0.Px3.p1.1.m1.1.1.2">𝑛</ci><cn type="integer" id="S2.SS2.SSS0.Px3.p1.1.m1.1.1.3.cmml" xref="S2.SS2.SSS0.Px3.p1.1.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px3.p1.1.m1.1c">n=3</annotation></semantics></math> and perform inference using GPT-4 with greedy decoding, retaining trajectories that yield correct answers. For questions where GPT-4 fails with greedy decoding, we apply nucleus sampling with a sample size of 10 and keep up to 4 valid trajectories per question.
Ultimately, we successfully annotate trajectories for 98.2% of GSM8k questions and 83.1% of MATH questions.
After filtering out invalid trajectories with tool-use errors or wrong answers, we obtain 16k annotations which constitute our dataset <span id="S2.SS2.SSS0.Px3.p1.1.1" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span>.
Table <a href="#S2.T1" title="Table 1 ‣ Trajectory Sampling ‣ 2.2 Collecting Interactive Tool-Use Trajectories ‣ 2 ToRA: Tool-Integrated Agents for Mathematical Reasoning ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a> compares <span id="S2.SS2.SSS0.Px3.p1.1.2" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span> with recently proposed mathematical reasoning datasets, while Table <a href="#A3.T6" title="Table 6 ‣ C.1 Accuracies of Closed-Source Models on MATH ‣ Appendix C Additional Experiments and Analysis ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">6</span></a> in the Appendix displays MATH annotation accuracy details.</p>
</div>
<figure id="S2.T1" class="ltx_table">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 1: </span>
Compared with mathematical reasoning datasets, <span id="S2.T1.5.1" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span> uniquely combines natural language rationales with program-based tool usage.
Note that <span id="S2.T1.6.2" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span> only employ questions from the original training set of MATH and GSM8k.
</figcaption>
<div id="S2.T1.2" class="ltx_inline-block ltx_align_center ltx_transformed_outer" style="width:433.6pt;height:80.8pt;vertical-align:-0.6pt;"><span class="ltx_transformed_inner" style="transform:translate(-123.8pt,22.9pt) scale(0.636596058795819,0.636596058795819) ;">
<table id="S2.T1.2.2" class="ltx_tabular ltx_align_middle">
<tr id="S2.T1.2.2.3" class="ltx_tr">
<td id="S2.T1.2.2.3.1" class="ltx_td ltx_align_left ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S2.T1.2.2.3.1.1" class="ltx_text ltx_font_bold">Methods</span></td>
<td id="S2.T1.2.2.3.2" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S2.T1.2.2.3.2.1" class="ltx_text ltx_font_bold">#Annotation</span></td>
<td id="S2.T1.2.2.3.3" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S2.T1.2.2.3.3.1" class="ltx_text ltx_font_bold">Tool</span></td>
<td id="S2.T1.2.2.3.4" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S2.T1.2.2.3.4.1" class="ltx_text ltx_font_bold">Interleaving</span></td>
<td id="S2.T1.2.2.3.5" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S2.T1.2.2.3.5.1" class="ltx_text ltx_font_bold">LLM Used</span></td>
<td id="S2.T1.2.2.3.6" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S2.T1.2.2.3.6.1" class="ltx_text ltx_font_bold">Source</span></td>
</tr>
<tr id="S2.T1.1.1.1" class="ltx_tr">
<td id="S2.T1.1.1.1.2" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">RFT <cite class="ltx_cite ltx_citemacro_citep">(Yuan et al., <a href="#bib.bib47" title="" class="ltx_ref">2023</a>)</cite>
</td>
<td id="S2.T1.1.1.1.1" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">
<math id="S2.T1.1.1.1.1.m1.1" class="ltx_Math" alttext="&gt;" display="inline"><semantics id="S2.T1.1.1.1.1.m1.1a"><mo id="S2.T1.1.1.1.1.m1.1.1" xref="S2.T1.1.1.1.1.m1.1.1.cmml">&gt;</mo><annotation-xml encoding="MathML-Content" id="S2.T1.1.1.1.1.m1.1b"><gt id="S2.T1.1.1.1.1.m1.1.1.cmml" xref="S2.T1.1.1.1.1.m1.1.1"></gt></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.1.1.1.1.m1.1c">&gt;</annotation></semantics></math>100k</td>
<td id="S2.T1.1.1.1.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S2.T1.1.1.1.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S2.T1.1.1.1.4" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S2.T1.1.1.1.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S2.T1.1.1.1.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">LLaMA-2</td>
<td id="S2.T1.1.1.1.6" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">GSM8k</td>
</tr>
<tr id="S2.T1.2.2.4" class="ltx_tr">
<td id="S2.T1.2.2.4.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Open-Platypus <cite class="ltx_cite ltx_citemacro_cite">Lee et al. (<a href="#bib.bib19" title="" class="ltx_ref">2023</a>)</cite>
</td>
<td id="S2.T1.2.2.4.2" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">25k</td>
<td id="S2.T1.2.2.4.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S2.T1.2.2.4.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S2.T1.2.2.4.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S2.T1.2.2.4.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S2.T1.2.2.4.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">GPT-4</td>
<td id="S2.T1.2.2.4.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">11 datasets with MATH</td>
</tr>
<tr id="S2.T1.2.2.2" class="ltx_tr">
<td id="S2.T1.2.2.2.2" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">WizardMath <cite class="ltx_cite ltx_citemacro_citep">(Luo et al., <a href="#bib.bib24" title="" class="ltx_ref">2023</a>)</cite>
</td>
<td id="S2.T1.2.2.2.1" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">
<math id="S2.T1.2.2.2.1.m1.1" class="ltx_Math" alttext="&gt;" display="inline"><semantics id="S2.T1.2.2.2.1.m1.1a"><mo id="S2.T1.2.2.2.1.m1.1.1" xref="S2.T1.2.2.2.1.m1.1.1.cmml">&gt;</mo><annotation-xml encoding="MathML-Content" id="S2.T1.2.2.2.1.m1.1b"><gt id="S2.T1.2.2.2.1.m1.1.1.cmml" xref="S2.T1.2.2.2.1.m1.1.1"></gt></annotation-xml><annotation encoding="application/x-tex" id="S2.T1.2.2.2.1.m1.1c">&gt;</annotation></semantics></math>96k</td>
<td id="S2.T1.2.2.2.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S2.T1.2.2.2.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S2.T1.2.2.2.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S2.T1.2.2.2.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S2.T1.2.2.2.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">ChatGPT</td>
<td id="S2.T1.2.2.2.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">MATH &amp; GSM8k</td>
</tr>
<tr id="S2.T1.2.2.5" class="ltx_tr">
<td id="S2.T1.2.2.5.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Lila <cite class="ltx_cite ltx_citemacro_citep">(Mishra et al., <a href="#bib.bib27" title="" class="ltx_ref">2022</a>)</cite>
</td>
<td id="S2.T1.2.2.5.2" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">134k</td>
<td id="S2.T1.2.2.5.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="S2.T1.2.2.5.3.1" class="ltx_text" style="color:#326400;">✓</span>(PoT)</td>
<td id="S2.T1.2.2.5.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S2.T1.2.2.5.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S2.T1.2.2.5.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S2.T1.2.2.5.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">20 datasets with MATH &amp; GSM8k</td>
</tr>
<tr id="S2.T1.2.2.6" class="ltx_tr">
<td id="S2.T1.2.2.6.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">MathInstruct <cite class="ltx_cite ltx_citemacro_citep">(Yue et al., <a href="#bib.bib48" title="" class="ltx_ref">2023</a>)</cite>
</td>
<td id="S2.T1.2.2.6.2" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">260k</td>
<td id="S2.T1.2.2.6.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="S2.T1.2.2.6.3.1" class="ltx_text" style="color:#326400;">✓</span>(PoT)</td>
<td id="S2.T1.2.2.6.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S2.T1.2.2.6.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S2.T1.2.2.6.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">GPT-4</td>
<td id="S2.T1.2.2.6.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">14 datasets with MATH &amp; GSM8k</td>
</tr>
<tr id="S2.T1.2.2.7" class="ltx_tr">
<td id="S2.T1.2.2.7.1" class="ltx_td ltx_align_left ltx_border_bb ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="S2.T1.2.2.7.1.1" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span> (ours)</td>
<td id="S2.T1.2.2.7.2" class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">16k</td>
<td id="S2.T1.2.2.7.3" class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S2.T1.2.2.7.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S2.T1.2.2.7.4" class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S2.T1.2.2.7.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S2.T1.2.2.7.5" class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">GPT-4</td>
<td id="S2.T1.2.2.7.6" class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">MATH &amp; GSM8k</td>
</tr>
</table>
</span></div>
</figure>
</section>
</section>
<section id="S2.SS3" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">2.3 </span>Training</h3>

<section id="S2.SS3.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Imitation Learning</h5>

<div id="S2.SS3.SSS0.Px1.p1" class="ltx_para ltx_noindent">
<p id="S2.SS3.SSS0.Px1.p1.2" class="ltx_p">We apply imitation learning on <span id="S2.SS3.SSS0.Px1.p1.2.1" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span> by minimizing negative log-likelihood loss on the trajectory <math id="S2.SS3.SSS0.Px1.p1.1.m1.1" class="ltx_Math" alttext="\tau" display="inline"><semantics id="S2.SS3.SSS0.Px1.p1.1.m1.1a"><mi id="S2.SS3.SSS0.Px1.p1.1.m1.1.1" xref="S2.SS3.SSS0.Px1.p1.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.1.m1.1b"><ci id="S2.SS3.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.1.m1.1c">\tau</annotation></semantics></math> conditioned on the problem <math id="S2.SS3.SSS0.Px1.p1.2.m2.1" class="ltx_Math" alttext="q" display="inline"><semantics id="S2.SS3.SSS0.Px1.p1.2.m2.1a"><mi id="S2.SS3.SSS0.Px1.p1.2.m2.1.1" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.2.m2.1b"><ci id="S2.SS3.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.2.m2.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.2.m2.1c">q</annotation></semantics></math>:</p>
<table id="S2.E4" class="ltx_equation ltx_eqn_table">

<tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="S2.E4.m1.9.9.9.9.9.9.1.2">𝑖</ci><cn type="integer" id="S2.E4.m1.9.9.9.9.9.9.1.3.cmml" xref="S2.E4.m1.9.9.9.9.9.9.1.3">1</cn></apply></apply><apply id="S2.E4.m1.10.10.10.10.10.10.1.cmml" xref="S2.E4.m1.10.10.10.10.10.10.1"><minus id="S2.E4.m1.10.10.10.10.10.10.1.1.cmml" xref="S2.E4.m1.10.10.10.10.10.10.1.1"></minus><ci id="S2.E4.m1.10.10.10.10.10.10.1.2.cmml" xref="S2.E4.m1.10.10.10.10.10.10.1.2">𝑛</ci><cn type="integer" id="S2.E4.m1.10.10.10.10.10.10.1.3.cmml" xref="S2.E4.m1.10.10.10.10.10.10.1.3">1</cn></apply></apply></apply></apply><apply id="S2.E4.m1.29.29.1.1.1.cmml" xref="S2.E4.m1.30.30.2"><times id="S2.E4.m1.29.29.1.1.1.2.cmml" xref="S2.E4.m1.30.30.2"></times><apply id="S2.E4.m1.29.29.1.1.1.3.cmml" xref="S2.E4.m1.30.30.2"><log id="S2.E4.m1.12.12.12.12.12.12.cmml" xref="S2.E4.m1.12.12.12.12.12.12"></log><apply id="S2.E4.m1.29.29.1.1.1.3.2.cmml" xref="S2.E4.m1.30.30.2"><csymbol cd="ambiguous" id="S2.E4.m1.29.29.1.1.1.3.2.1.cmml" xref="S2.E4.m1.30.30.2">subscript</csymbol><ci id="S2.E4.m1.13.13.13.13.13.13.cmml" xref="S2.E4.m1.13.13.13.13.13.13">ℙ</ci><ci id="S2.E4.m1.14.14.14.14.14.14.1.cmml" xref="S2.E4.m1.14.14.14.14.14.14.1">ℳ</ci></apply></apply><apply id="S2.E4.m1.29.29.1.1.1.1.1.1.cmml" xref="S2.E4.m1.30.30.2"><csymbol cd="latexml" id="S2.E4.m1.20.20.20.20.20.20.cmml" xref="S2.E4.m1.20.20.20.20.20.20">conditional</csymbol><apply id="S2.E4.m1.29.29.1.1.1.1.1.1.3.cmml" xref="S2.E4.m1.30.30.2"><times id="S2.E4.m1.29.29.1.1.1.1.1.1.3.1.cmml" xref="S2.E4.m1.30.30.2"></times><apply id="S2.E4.m1.29.29.1.1.1.1.1.1.3.2.cmml" xref="S2.E4.m1.30.30.2"><csymbol cd="ambiguous" id="S2.E4.m1.29.29.1.1.1.1.1.1.3.2.1.cmml" xref="S2.E4.m1.30.30.2">subscript</csymbol><ci id="S2.E4.m1.16.16.16.16.16.16.cmml" xref="S2.E4.m1.16.16.16.16.16.16">𝑟</ci><apply id="S2.E4.m1.17.17.17.17.17.17.1.cmml" xref="S2.E4.m1.17.17.17.17.17.17.1"><plus id="S2.E4.m1.17.17.17.17.17.17.1.1.cmml" xref="S2.E4.m1.17.17.17.17.17.17.1.1"></plus><ci id="S2.E4.m1.17.17.17.17.17.17.1.2.cmml" xref="S2.E4.m1.17.17.17.17.17.17.1.2">𝑖</ci><cn type="integer" id="S2.E4.m1.17.17.17.17.17.17.1.3.cmml" xref="S2.E4.m1.17.17.17.17.17.17.1.3">1</cn></apply></apply><apply id="S2.E4.m1.29.29.1.1.1.1.1.1.3.3.cmml" xref="S2.E4.m1.30.30.2"><csymbol cd="ambiguous" id="S2.E4.m1.29.29.1.1.1.1.1.1.3.3.1.cmml" xref="S2.E4.m1.30.30.2">subscript</csymbol><ci id="S2.E4.m1.18.18.18.18.18.18.cmml" xref="S2.E4.m1.18.18.18.18.18.18">𝑎</ci><apply id="S2.E4.m1.19.19.19.19.19.19.1.cmml" xref="S2.E4.m1.19.19.19.19.19.19.1"><plus id="S2.E4.m1.19.19.19.19.19.19.1.1.cmml" xref="S2.E4.m1.19.19.19.19.19.19.1.1"></plus><ci id="S2.E4.m1.19.19.19.19.19.19.1.2.cmml" xref="S2.E4.m1.19.19.19.19.19.19.1.2">𝑖</ci><cn type="integer" id="S2.E4.m1.19.19.19.19.19.19.1.3.cmml" xref="S2.E4.m1.19.19.19.19.19.19.1.3">1</cn></apply></apply></apply><list id="S2.E4.m1.29.29.1.1.1.1.1.1.1.2.cmml" xref="S2.E4.m1.30.30.2"><ci id="S2.E4.m1.21.21.21.21.21.21.cmml" xref="S2.E4.m1.21.21.21.21.21.21">𝑞</ci><apply id="S2.E4.m1.29.29.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E4.m1.30.30.2"><times id="S2.E4.m1.29.29.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E4.m1.30.30.2"></times><apply id="S2.E4.m1.29.29.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.E4.m1.30.30.2"><csymbol cd="ambiguous" id="S2.E4.m1.29.29.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S2.E4.m1.30.30.2">subscript</csymbol><ci id="S2.E4.m1.23.23.23.23.23.23.cmml" xref="S2.E4.m1.23.23.23.23.23.23">𝑟</ci><cn type="integer" id="S2.E4.m1.24.24.24.24.24.24.1.cmml" xref="S2.E4.m1.24.24.24.24.24.24.1">1</cn></apply><ci id="S2.E4.m1.25.25.25.25.25.25.cmml" xref="S2.E4.m1.25.25.25.25.25.25">…</ci><apply id="S2.E4.m1.29.29.1.1.1.1.1.1.1.1.1.4.cmml" xref="S2.E4.m1.30.30.2"><csymbol cd="ambiguous" id="S2.E4.m1.29.29.1.1.1.1.1.1.1.1.1.4.1.cmml" xref="S2.E4.m1.30.30.2">subscript</csymbol><ci id="S2.E4.m1.26.26.26.26.26.26.cmml" xref="S2.E4.m1.26.26.26.26.26.26">𝑜</ci><ci id="S2.E4.m1.27.27.27.27.27.27.1.cmml" xref="S2.E4.m1.27.27.27.27.27.27.1">𝑖</ci></apply></apply></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.m1.30c">\begin{split}\mathcal{M}=\arg\min_{\mathcal{M}}\sum_{q,\tau}\sum_{i=1}^{n-1}-\log\mathbb{P}_{\mathcal{M}}(r_{i+1}a_{i+1}|q,r_{1}...o_{i})\end{split}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td>
</tr></tbody>
</table>
<p id="S2.SS3.SSS0.Px1.p1.4" class="ltx_p">where <math id="S2.SS3.SSS0.Px1.p1.3.m1.1" class="ltx_Math" alttext="\mathcal{M}" display="inline"><semantics id="S2.SS3.SSS0.Px1.p1.3.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.SSS0.Px1.p1.3.m1.1.1" xref="S2.SS3.SSS0.Px1.p1.3.m1.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.3.m1.1b"><ci id="S2.SS3.SSS0.Px1.p1.3.m1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.3.m1.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.3.m1.1c">\mathcal{M}</annotation></semantics></math> is the resulting model.
After imitation learning, we can simply apply the same procedure in Algorithm <a href="#alg1" title="Algorithm 1 ‣ 2.2 Collecting Interactive Tool-Use Trajectories ‣ 2 ToRA: Tool-Integrated Agents for Mathematical Reasoning ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a> by setting prompt to empty <math id="S2.SS3.SSS0.Px1.p1.4.m2.1" class="ltx_Math" alttext='p=\text{""}' display="inline"><semantics id="S2.SS3.SSS0.Px1.p1.4.m2.1a"><mrow id="S2.SS3.SSS0.Px1.p1.4.m2.1.1" xref="S2.SS3.SSS0.Px1.p1.4.m2.1.1.cmml"><mi id="S2.SS3.SSS0.Px1.p1.4.m2.1.1.2" xref="S2.SS3.SSS0.Px1.p1.4.m2.1.1.2.cmml">p</mi><mo id="S2.SS3.SSS0.Px1.p1.4.m2.1.1.1" xref="S2.SS3.SSS0.Px1.p1.4.m2.1.1.1.cmml">=</mo><mtext id="S2.SS3.SSS0.Px1.p1.4.m2.1.1.3" xref="S2.SS3.SSS0.Px1.p1.4.m2.1.1.3a.cmml">""</mtext></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p1.4.m2.1b"><apply id="S2.SS3.SSS0.Px1.p1.4.m2.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.4.m2.1.1"><eq id="S2.SS3.SSS0.Px1.p1.4.m2.1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p1.4.m2.1.1.1"></eq><ci id="S2.SS3.SSS0.Px1.p1.4.m2.1.1.2.cmml" xref="S2.SS3.SSS0.Px1.p1.4.m2.1.1.2">𝑝</ci><ci id="S2.SS3.SSS0.Px1.p1.4.m2.1.1.3a.cmml" xref="S2.SS3.SSS0.Px1.p1.4.m2.1.1.3"><mtext id="S2.SS3.SSS0.Px1.p1.4.m2.1.1.3.cmml" xref="S2.SS3.SSS0.Px1.p1.4.m2.1.1.3">""</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p1.4.m2.1c">p=\text{""}</annotation></semantics></math> for inference.
Imitation learning leads to state-of-the-art mathematical reasoning performance despite the small scale of <span id="S2.SS3.SSS0.Px1.p1.4.1" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span>.</p>
</div>
</section>
<section id="S2.SS3.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Output Space Shaping</h5>

<div id="S2.SS3.SSS0.Px2.p1" class="ltx_para ltx_noindent">
<p id="S2.SS3.SSS0.Px2.p1.1" class="ltx_p">For each question, <span id="S2.SS3.SSS0.Px2.p1.1.1" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span> mostly demonstrates only one valid interactive tool-use trajectory, which may restrict a model’s output space, rendering it inflexible in exploring plausible trajectories during testing.
We therefore propose <em id="S2.SS3.SSS0.Px2.p1.1.2" class="ltx_emph ltx_font_italic">output space shaping</em> in order to encourage the diversity of plausible reasoning steps and reduce improper tool-use behavior.</p>
</div>
<div id="S2.SS3.SSS0.Px2.p2" class="ltx_para ltx_noindent">
<p id="S2.SS3.SSS0.Px2.p2.10" class="ltx_p">To explore diverse valid trajectories, we apply nucleus sampling to imitation learning models <math id="S2.SS3.SSS0.Px2.p2.1.m1.1" class="ltx_Math" alttext="\mathcal{M}" display="inline"><semantics id="S2.SS3.SSS0.Px2.p2.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.SSS0.Px2.p2.1.m1.1.1" xref="S2.SS3.SSS0.Px2.p2.1.m1.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p2.1.m1.1b"><ci id="S2.SS3.SSS0.Px2.p2.1.m1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.1.m1.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p2.1.m1.1c">\mathcal{M}</annotation></semantics></math> to sample 64 trajectories per training question <math id="S2.SS3.SSS0.Px2.p2.2.m2.1" class="ltx_Math" alttext="q" display="inline"><semantics id="S2.SS3.SSS0.Px2.p2.2.m2.1a"><mi id="S2.SS3.SSS0.Px2.p2.2.m2.1.1" xref="S2.SS3.SSS0.Px2.p2.2.m2.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p2.2.m2.1b"><ci id="S2.SS3.SSS0.Px2.p2.2.m2.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.2.m2.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p2.2.m2.1c">q</annotation></semantics></math>, following the inference procedure in Section <a href="#S2.SS2" title="2.2 Collecting Interactive Tool-Use Trajectories ‣ 2 ToRA: Tool-Integrated Agents for Mathematical Reasoning ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2.2</span></a>.
We retain valid trajectories with correct answers and no tool-use errors.
As many samples are duplicates, to further improve diversity and in an attempt to correct models’ improper behavior, we seek to leverage invalid trajectories as well.
We observe that trajectories with wrong answers are mostly incorrect halfway <cite class="ltx_cite ltx_citemacro_citep">(Li et al., <a href="#bib.bib20" title="" class="ltx_ref">2023</a>)</cite>, and the preceding reasoning is still plausible; in other words, we can obtain valid trajectories by correcting the subsequent portions.
Specifically, a wrong trajectory <math id="S2.SS3.SSS0.Px2.p2.3.m3.1" class="ltx_Math" alttext="\widetilde{\tau}" display="inline"><semantics id="S2.SS3.SSS0.Px2.p2.3.m3.1a"><mover accent="true" id="S2.SS3.SSS0.Px2.p2.3.m3.1.1" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1.cmml"><mi id="S2.SS3.SSS0.Px2.p2.3.m3.1.1.2" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1.2.cmml">τ</mi><mo id="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p2.3.m3.1b"><apply id="S2.SS3.SSS0.Px2.p2.3.m3.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1"><ci id="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1.1">~</ci><ci id="S2.SS3.SSS0.Px2.p2.3.m3.1.1.2.cmml" xref="S2.SS3.SSS0.Px2.p2.3.m3.1.1.2">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p2.3.m3.1c">\widetilde{\tau}</annotation></semantics></math>, when written in text, can be represented as a sequence of lines separated by line breaks, i.e., <math id="S2.SS3.SSS0.Px2.p2.4.m4.1" class="ltx_Math" alttext="\widetilde{\tau}=l_{1}...l_{m}" display="inline"><semantics id="S2.SS3.SSS0.Px2.p2.4.m4.1a"><mrow id="S2.SS3.SSS0.Px2.p2.4.m4.1.1" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.cmml"><mover accent="true" id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.2" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.2.cmml"><mi id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.2.2" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.2.2.cmml">τ</mi><mo id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.2.1" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.2.1.cmml">~</mo></mover><mo id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.1" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.1.cmml">=</mo><mrow id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.cmml"><msub id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.2" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.2.cmml"><mi id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.2.2" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.2.2.cmml">l</mi><mn id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.2.3" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.2.3.cmml">1</mn></msub><mo lspace="0em" rspace="0em" id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.1" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.1.cmml">​</mo><mi mathvariant="normal" id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.3" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.3.cmml">…</mi><mo lspace="0em" rspace="0em" id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.1a" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.1.cmml">​</mo><msub id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.4" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.4.cmml"><mi id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.4.2" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.4.2.cmml">l</mi><mi id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.4.3" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.4.3.cmml">m</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p2.4.m4.1b"><apply id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1"><eq id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.1"></eq><apply id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.2.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.2"><ci id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.2.1.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.2.1">~</ci><ci id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.2.2.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.2.2">𝜏</ci></apply><apply id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3"><times id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.1"></times><apply id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.2.1.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.2">subscript</csymbol><ci id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.2.2.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.2.2">𝑙</ci><cn type="integer" id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.2.3.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.2.3">1</cn></apply><ci id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.3.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.3">…</ci><apply id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.4.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.4"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.4.1.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.4">subscript</csymbol><ci id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.4.2.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.4.2">𝑙</ci><ci id="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.4.3.cmml" xref="S2.SS3.SSS0.Px2.p2.4.m4.1.1.3.4.3">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p2.4.m4.1c">\widetilde{\tau}=l_{1}...l_{m}</annotation></semantics></math>, where <math id="S2.SS3.SSS0.Px2.p2.5.m5.1" class="ltx_Math" alttext="m" display="inline"><semantics id="S2.SS3.SSS0.Px2.p2.5.m5.1a"><mi id="S2.SS3.SSS0.Px2.p2.5.m5.1.1" xref="S2.SS3.SSS0.Px2.p2.5.m5.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p2.5.m5.1b"><ci id="S2.SS3.SSS0.Px2.p2.5.m5.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.5.m5.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p2.5.m5.1c">m</annotation></semantics></math> is the total number of lines in <math id="S2.SS3.SSS0.Px2.p2.6.m6.1" class="ltx_Math" alttext="\widetilde{\tau}" display="inline"><semantics id="S2.SS3.SSS0.Px2.p2.6.m6.1a"><mover accent="true" id="S2.SS3.SSS0.Px2.p2.6.m6.1.1" xref="S2.SS3.SSS0.Px2.p2.6.m6.1.1.cmml"><mi id="S2.SS3.SSS0.Px2.p2.6.m6.1.1.2" xref="S2.SS3.SSS0.Px2.p2.6.m6.1.1.2.cmml">τ</mi><mo id="S2.SS3.SSS0.Px2.p2.6.m6.1.1.1" xref="S2.SS3.SSS0.Px2.p2.6.m6.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p2.6.m6.1b"><apply id="S2.SS3.SSS0.Px2.p2.6.m6.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.6.m6.1.1"><ci id="S2.SS3.SSS0.Px2.p2.6.m6.1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.6.m6.1.1.1">~</ci><ci id="S2.SS3.SSS0.Px2.p2.6.m6.1.1.2.cmml" xref="S2.SS3.SSS0.Px2.p2.6.m6.1.1.2">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p2.6.m6.1c">\widetilde{\tau}</annotation></semantics></math>.
We enumerate possible preceding portions of wrong trajectories, i.e., <math id="S2.SS3.SSS0.Px2.p2.7.m7.1" class="ltx_math_unparsed" alttext="\widetilde{\tau}[:j]=l_{1}...l_{j}" display="inline"><semantics id="S2.SS3.SSS0.Px2.p2.7.m7.1a"><mrow id="S2.SS3.SSS0.Px2.p2.7.m7.1b"><mover accent="true" id="S2.SS3.SSS0.Px2.p2.7.m7.1.1"><mi id="S2.SS3.SSS0.Px2.p2.7.m7.1.1.2">τ</mi><mo id="S2.SS3.SSS0.Px2.p2.7.m7.1.1.1">~</mo></mover><mrow id="S2.SS3.SSS0.Px2.p2.7.m7.1.2"><mo stretchy="false" id="S2.SS3.SSS0.Px2.p2.7.m7.1.2.1">[</mo><mo rspace="0.278em" id="S2.SS3.SSS0.Px2.p2.7.m7.1.2.2">:</mo><mi id="S2.SS3.SSS0.Px2.p2.7.m7.1.2.3">j</mi><mo stretchy="false" id="S2.SS3.SSS0.Px2.p2.7.m7.1.2.4">]</mo></mrow><mo id="S2.SS3.SSS0.Px2.p2.7.m7.1.3">=</mo><msub id="S2.SS3.SSS0.Px2.p2.7.m7.1.4"><mi id="S2.SS3.SSS0.Px2.p2.7.m7.1.4.2">l</mi><mn id="S2.SS3.SSS0.Px2.p2.7.m7.1.4.3">1</mn></msub><mi mathvariant="normal" id="S2.SS3.SSS0.Px2.p2.7.m7.1.5">…</mi><msub id="S2.SS3.SSS0.Px2.p2.7.m7.1.6"><mi id="S2.SS3.SSS0.Px2.p2.7.m7.1.6.2">l</mi><mi id="S2.SS3.SSS0.Px2.p2.7.m7.1.6.3">j</mi></msub></mrow><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p2.7.m7.1c">\widetilde{\tau}[:j]=l_{1}...l_{j}</annotation></semantics></math>, and leverage a teacher model <math id="S2.SS3.SSS0.Px2.p2.8.m8.1" class="ltx_Math" alttext="\mathcal{M}^{\prime}" display="inline"><semantics id="S2.SS3.SSS0.Px2.p2.8.m8.1a"><msup id="S2.SS3.SSS0.Px2.p2.8.m8.1.1" xref="S2.SS3.SSS0.Px2.p2.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.SSS0.Px2.p2.8.m8.1.1.2" xref="S2.SS3.SSS0.Px2.p2.8.m8.1.1.2.cmml">ℳ</mi><mo id="S2.SS3.SSS0.Px2.p2.8.m8.1.1.3" xref="S2.SS3.SSS0.Px2.p2.8.m8.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p2.8.m8.1b"><apply id="S2.SS3.SSS0.Px2.p2.8.m8.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px2.p2.8.m8.1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.8.m8.1.1">superscript</csymbol><ci id="S2.SS3.SSS0.Px2.p2.8.m8.1.1.2.cmml" xref="S2.SS3.SSS0.Px2.p2.8.m8.1.1.2">ℳ</ci><ci id="S2.SS3.SSS0.Px2.p2.8.m8.1.1.3.cmml" xref="S2.SS3.SSS0.Px2.p2.8.m8.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p2.8.m8.1c">\mathcal{M}^{\prime}</annotation></semantics></math> to complete the subsequent steps with greedy decoding:
<math id="S2.SS3.SSS0.Px2.p2.9.m9.1" class="ltx_math_unparsed" alttext="\tau\leftarrow\mathbb{P}_{\mathcal{M}^{\prime}}(\cdot|q\oplus\widetilde{\tau}[:j])" display="inline"><semantics id="S2.SS3.SSS0.Px2.p2.9.m9.1a"><mrow id="S2.SS3.SSS0.Px2.p2.9.m9.1b"><mi id="S2.SS3.SSS0.Px2.p2.9.m9.1.1">τ</mi><mo stretchy="false" id="S2.SS3.SSS0.Px2.p2.9.m9.1.2">←</mo><msub id="S2.SS3.SSS0.Px2.p2.9.m9.1.3"><mi id="S2.SS3.SSS0.Px2.p2.9.m9.1.3.2">ℙ</mi><msup id="S2.SS3.SSS0.Px2.p2.9.m9.1.3.3"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.SSS0.Px2.p2.9.m9.1.3.3.2">ℳ</mi><mo id="S2.SS3.SSS0.Px2.p2.9.m9.1.3.3.3">′</mo></msup></msub><mrow id="S2.SS3.SSS0.Px2.p2.9.m9.1.4"><mo stretchy="false" id="S2.SS3.SSS0.Px2.p2.9.m9.1.4.1">(</mo><mo lspace="0em" rspace="0em" id="S2.SS3.SSS0.Px2.p2.9.m9.1.4.2">⋅</mo><mo fence="false" rspace="0.167em" stretchy="false" id="S2.SS3.SSS0.Px2.p2.9.m9.1.4.3">|</mo><mi id="S2.SS3.SSS0.Px2.p2.9.m9.1.4.4">q</mi><mo id="S2.SS3.SSS0.Px2.p2.9.m9.1.4.5">⊕</mo><mover accent="true" id="S2.SS3.SSS0.Px2.p2.9.m9.1.4.6"><mi id="S2.SS3.SSS0.Px2.p2.9.m9.1.4.6.2">τ</mi><mo id="S2.SS3.SSS0.Px2.p2.9.m9.1.4.6.1">~</mo></mover><mrow id="S2.SS3.SSS0.Px2.p2.9.m9.1.4.7"><mo stretchy="false" id="S2.SS3.SSS0.Px2.p2.9.m9.1.4.7.1">[</mo><mo rspace="0.278em" id="S2.SS3.SSS0.Px2.p2.9.m9.1.4.7.2">:</mo><mi id="S2.SS3.SSS0.Px2.p2.9.m9.1.4.7.3">j</mi><mo stretchy="false" id="S2.SS3.SSS0.Px2.p2.9.m9.1.4.7.4">]</mo></mrow><mo stretchy="false" id="S2.SS3.SSS0.Px2.p2.9.m9.1.4.8">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p2.9.m9.1c">\tau\leftarrow\mathbb{P}_{\mathcal{M}^{\prime}}(\cdot|q\oplus\widetilde{\tau}[:j])</annotation></semantics></math> where we abuse the notation <math id="S2.SS3.SSS0.Px2.p2.10.m10.1" class="ltx_Math" alttext="\mathbb{P}_{\mathcal{M}^{\prime}}(\cdot)" display="inline"><semantics id="S2.SS3.SSS0.Px2.p2.10.m10.1a"><mrow id="S2.SS3.SSS0.Px2.p2.10.m10.1.2" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.cmml"><msub id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.cmml"><mi id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.2" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.2.cmml">ℙ</mi><msup id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.3" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.3.2" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.3.2.cmml">ℳ</mi><mo id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.3.3" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.3.3.cmml">′</mo></msup></msub><mo lspace="0em" rspace="0em" id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.1" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.1.cmml">​</mo><mrow id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.3.2" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.cmml"><mo stretchy="false" id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.3.2.1" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.cmml">(</mo><mo lspace="0em" rspace="0em" id="S2.SS3.SSS0.Px2.p2.10.m10.1.1" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.1.cmml">⋅</mo><mo stretchy="false" id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.3.2.2" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p2.10.m10.1b"><apply id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.cmml" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2"><times id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.1.cmml" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.1"></times><apply id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.cmml" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.1.cmml" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2">subscript</csymbol><ci id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.2.cmml" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.2">ℙ</ci><apply id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.3.cmml" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.3.1.cmml" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.3">superscript</csymbol><ci id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.3.2.cmml" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.3.2">ℳ</ci><ci id="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.3.3.cmml" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.2.2.3.3">′</ci></apply></apply><ci id="S2.SS3.SSS0.Px2.p2.10.m10.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.10.m10.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p2.10.m10.1c">\mathbb{P}_{\mathcal{M}^{\prime}}(\cdot)</annotation></semantics></math> to denote the interactive tool use process following Section <a href="#S2.SS2" title="2.2 Collecting Interactive Tool-Use Trajectories ‣ 2 ToRA: Tool-Integrated Agents for Mathematical Reasoning ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2.2</span></a>.
Finally, corrected trajectories as well as valid trajectory samples will be used for model training, thereby shaping the output space.</p>
</div>
<div id="S2.SS3.SSS0.Px2.p3" class="ltx_para ltx_noindent">
<p id="S2.SS3.SSS0.Px2.p3.1" class="ltx_p">In our experiments, we always use CodeLLaMA-34B trained on <span id="S2.SS3.SSS0.Px2.p3.1.1" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span> as the teacher model, and apply sampling with the CodeLLaMA series (ranging from 7B to 34B, with imitation learning).
We obtain a total of 233k distinct valid trajectory samples and 69k corrected ones.
From this combined dataset, we randomly select up to 4 trajectories per GSM8k and MATH problem, merge them with <span id="S2.SS3.SSS0.Px2.p3.1.2" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span>, and then train all <span id="S2.SS3.SSS0.Px2.p3.1.3" class="ltx_text ltx_font_smallcaps">ToRA</span> models on the resulting 69k annotations.</p>
</div>
</section>
</section>
</section>
<section id="S3" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">3 </span>Experiments</h2>

<section id="S3.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">3.1 </span>Implementation Details</h3>

<div id="S3.SS1.p1" class="ltx_para ltx_noindent">
<p id="S3.SS1.p1.1" class="ltx_p">We fine-tuned LLaMA-2 <cite class="ltx_cite ltx_citemacro_citep">(Touvron et al., <a href="#bib.bib44" title="" class="ltx_ref">2023b</a>)</cite> and CodeLLaMA <cite class="ltx_cite ltx_citemacro_citep">(Rozière et al., <a href="#bib.bib37" title="" class="ltx_ref">2023</a>)</cite> series (ranging from 7B to 70B) using <span id="S3.SS1.p1.1.1" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span> with output space shaping, yielding the <span id="S3.SS1.p1.1.2" class="ltx_text ltx_font_smallcaps">ToRA</span> and <span id="S3.SS1.p1.1.3" class="ltx_text ltx_font_smallcaps">ToRA-Code</span> series respectively.
We used a learning rate of 2e-5 by default except that we used 1e-5 for the 34B and 70B models.
We set the global batch size to 128 and used a linear scheduler with a 3% warm-up period for 3 epochs.
We trained all models with <span id="S3.SS1.p1.1.4" class="ltx_text ltx_font_italic">DeepSpeed ZeRO Stage3</span> <cite class="ltx_cite ltx_citemacro_citep">(Rajbhandari et al., <a href="#bib.bib36" title="" class="ltx_ref">2021</a>)</cite> and <span id="S3.SS1.p1.1.5" class="ltx_text ltx_font_italic">Flash-Attention 2</span> <cite class="ltx_cite ltx_citemacro_citep">(Dao, <a href="#bib.bib7" title="" class="ltx_ref">2023</a>)</cite>.
We used greedy decoding for all results, with the maximum sequence length set to 2,048 and the maximum number of tool executions set to 3.</p>
</div>
</section>
<section id="S3.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">3.2 </span>Evaluation Setup</h3>

<div id="S3.SS2.p1" class="ltx_para ltx_noindent">
<p id="S3.SS2.p1.1" class="ltx_p"><span id="S3.SS2.p1.1.1" class="ltx_text ltx_font_bold">Datasets</span> We evaluated models on GSM8k <cite class="ltx_cite ltx_citemacro_citep">(Cobbe et al., <a href="#bib.bib6" title="" class="ltx_ref">2021</a>)</cite> and MATH <cite class="ltx_cite ltx_citemacro_citep">(Hendrycks et al., <a href="#bib.bib12" title="" class="ltx_ref">2021</a>)</cite>, along with 8 out-of-distribution datasets, namely GSM-Hard <cite class="ltx_cite ltx_citemacro_citep">(Gao et al., <a href="#bib.bib10" title="" class="ltx_ref">2022</a>)</cite>, SVAMP <cite class="ltx_cite ltx_citemacro_citep">(Patel et al., <a href="#bib.bib32" title="" class="ltx_ref">2021</a>)</cite>, ASDIV <cite class="ltx_cite ltx_citemacro_citep">(Miao et al., <a href="#bib.bib26" title="" class="ltx_ref">2020</a>)</cite>, TabMWP <cite class="ltx_cite ltx_citemacro_citep">(Lu et al., <a href="#bib.bib23" title="" class="ltx_ref">2023</a>)</cite>, SingleEQ, SingleOP, AddSub, and MultiArith <cite class="ltx_cite ltx_citemacro_citep">(Koncel-Kedziorski et al., <a href="#bib.bib18" title="" class="ltx_ref">2016</a>)</cite>, as illustrated in Table <a href="#A2.T5" title="Table 5 ‣ Appendix B Evaluation Datasets ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">5</span></a> in Appendix.
The 10 assorted datasets collectively encompass mathematical problems spanning basic arithmetic to competition level, covering middle and high school curricula and various mathematical domains.
The problem formats comprise tabular-based, free-form, and multiple-choice questions, ensuring a thorough assessment of the model’s mathematical reasoning aptitude.</p>
</div>
<div id="S3.SS2.p2" class="ltx_para ltx_noindent">
<p id="S3.SS2.p2.1" class="ltx_p"><span id="S3.SS2.p2.1.1" class="ltx_text ltx_font_bold">Metrics</span>
We report accuracies of predicted answers.
Following <cite class="ltx_cite ltx_citemacro_citet">Lightman et al. (<a href="#bib.bib22" title="" class="ltx_ref">2023</a>)</cite>, we round numerical values and use <span id="S3.SS2.p2.1.2" class="ltx_text ltx_font_typewriter">sympy</span> <span id="footnote2" class="ltx_note ltx_role_footnote"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_tag ltx_tag_note">2</span><a target="_blank" href="https://www.sympy.org" title="" class="ltx_ref ltx_url ltx_font_typewriter">https://www.sympy.org</a></span></span></span> for parsing expressions.
Since the SingleEQ, SingleOP, AddSub, and MultiArith datasets focus on different aspects of basic arithmetic, we report their average results under the collective term MAWPS <cite class="ltx_cite ltx_citemacro_citep">(Koncel-Kedziorski et al., <a href="#bib.bib18" title="" class="ltx_ref">2016</a>)</cite> for all methods.</p>
</div>
</section>
<section id="S3.SS3" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">3.3 </span>Baselines</h3>

<div id="S3.SS3.p1" class="ltx_para ltx_noindent">
<p id="S3.SS3.p1.1" class="ltx_p"><span id="S3.SS3.p1.1.1" class="ltx_text ltx_font_bold">Proprietary Models</span>
We present results from an array of SoTA LLMs, such as OpenAI’s GPT-4, ChatGPT (<span id="S3.SS3.p1.1.2" class="ltx_text ltx_font_typewriter">gpt-3.5-turbo</span>), Google’s PaLM-2, and Anthropic’s Claude-2. By default, we report CoT prompting results, and include PAL <cite class="ltx_cite ltx_citemacro_citep">(Gao et al., <a href="#bib.bib10" title="" class="ltx_ref">2022</a>)</cite> prompting results for selected models.</p>
</div>
<div id="S3.SS3.p2" class="ltx_para ltx_noindent">
<p id="S3.SS3.p2.1" class="ltx_p"><span id="S3.SS3.p2.1.1" class="ltx_text ltx_font_bold">Open-Source Models</span>
<span id="S3.SS3.p2.1.2" class="ltx_text ltx_font_italic">Base models</span> comprise LLaMA-2 and CodeLLaMA with CoT and PAL prompting. <span id="S3.SS3.p2.1.3" class="ltx_text ltx_font_italic">Supervised Fine-Tuning (SFT)</span> employs CoT rationales from the original GSM8k and MATH dataset (15k samples) for fine-tuning. <span id="S3.SS3.p2.1.4" class="ltx_text ltx_font_italic">Rejection sampling Fine-Tuning (RFT)</span> leverages multiple models to generate diverse reasoning paths for fine-tuning <cite class="ltx_cite ltx_citemacro_citep">(Yuan et al., <a href="#bib.bib47" title="" class="ltx_ref">2023</a>)</cite>. <span id="S3.SS3.p2.1.5" class="ltx_text ltx_font_italic">WizardMath</span> augments data using ChatGPT, and conducts SFT and RLHF. <span id="S3.SS3.p2.1.6" class="ltx_text ltx_font_italic">Platypus-2</span>, the top model on the LLM Leaderboard <span id="footnote3" class="ltx_note ltx_role_footnote"><sup class="ltx_note_mark">3</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">3</sup><span class="ltx_tag ltx_tag_note">3</span><a target="_blank" href="https://huggingface.co/spaces/HuggingFaceH4/open_llm_leaderboard" title="" class="ltx_ref ltx_url ltx_font_typewriter">https://huggingface.co/spaces/HuggingFaceH4/open_llm_leaderboard</a></span></span></span>, is fine-tuned with Open-Platypus reasoning datasets <cite class="ltx_cite ltx_citemacro_citep">(Lee et al., <a href="#bib.bib19" title="" class="ltx_ref">2023</a>)</cite>.
We also compare <span id="S3.SS3.p2.1.7" class="ltx_text ltx_font_smallcaps">ToRA</span> with Toolformer <cite class="ltx_cite ltx_citemacro_citep">(Schick et al., <a href="#bib.bib38" title="" class="ltx_ref">2023</a>)</cite> which is a model trained to utilize calculators.</p>
</div>
<figure id="S3.T2" class="ltx_table">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 2: </span>Results on 10 mathematical reasoning tasks. MAWPS results are averaged over four tasks: Singleeq, Singleop, Addsub, and MultArith. Vanilla models are tested with CoT. The best results in each section are in <span id="S3.T2.20.1" class="ltx_text" style="background-color:#D2DCFA;">blue</span>, the second-best results are underlined, while the results of our best model are bolded. <sup id="S3.T2.21.2" class="ltx_sup">∗</sup> ZS: Zero-shot inference without demonstrations.
</figcaption>
<div id="S3.T2.17" class="ltx_inline-block ltx_align_center ltx_transformed_outer" style="width:433.6pt;height:516.8pt;vertical-align:-0.8pt;"><span class="ltx_transformed_inner" style="transform:translate(-63.0pt,75.0pt) scale(0.774785022165755,0.774785022165755) ;">
<table id="S3.T2.17.15" class="ltx_tabular ltx_align_middle">
<tr id="S3.T2.3.1.1" class="ltx_tr">
<td id="S3.T2.3.1.1.2" class="ltx_td ltx_align_left ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.3.1.1.2.1" class="ltx_text ltx_font_bold">Model</span></td>
<td id="S3.T2.3.1.1.3" class="ltx_td ltx_align_right ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.3.1.1.3.1" class="ltx_text ltx_font_bold">Size</span></td>
<td id="S3.T2.3.1.1.4" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.3.1.1.4.1" class="ltx_text ltx_font_bold">Tools</span></td>
<td id="S3.T2.3.1.1.1" class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.3.1.1.1.1" class="ltx_text ltx_font_bold">ZS<sup id="S3.T2.3.1.1.1.1.1" class="ltx_sup"><span id="S3.T2.3.1.1.1.1.1.1" class="ltx_text ltx_font_medium">∗</span></sup></span></td>
<td id="S3.T2.3.1.1.5" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.3.1.1.5.1" class="ltx_text ltx_font_bold">GSM8k</span></td>
<td id="S3.T2.3.1.1.6" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.3.1.1.6.1" class="ltx_text ltx_font_bold">MATH</span></td>
<td id="S3.T2.3.1.1.7" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.3.1.1.7.1" class="ltx_text ltx_font_bold">GSM-Hard</span></td>
<td id="S3.T2.3.1.1.8" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.3.1.1.8.1" class="ltx_text ltx_font_bold">SVAMP</span></td>
<td id="S3.T2.3.1.1.9" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.3.1.1.9.1" class="ltx_text ltx_font_bold">TabMWP</span></td>
<td id="S3.T2.3.1.1.10" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.3.1.1.10.1" class="ltx_text ltx_font_bold">ASDiv</span></td>
<td id="S3.T2.3.1.1.11" class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.3.1.1.11.1" class="ltx_text ltx_font_bold">MAWPS</span></td>
<td id="S3.T2.3.1.1.12" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;" rowspan="2"><span id="S3.T2.3.1.1.12.1" class="ltx_text ltx_font_bold">AVG</span></td>
</tr>
<tr id="S3.T2.17.15.16" class="ltx_tr">
<td id="S3.T2.17.15.16.1" class="ltx_td ltx_align_left ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;" colspan="4"><span id="S3.T2.17.15.16.1.1" class="ltx_text ltx_font_bold">Used for training?</span></td>
<td id="S3.T2.17.15.16.2" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.16.2.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.17.15.16.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.16.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.17.15.16.4" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.16.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.16.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.16.5.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.16.6" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.16.6.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.16.7" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.16.7.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.16.8" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.16.8.1" class="ltx_text" style="color:#C80000;">✗</span></td>
</tr>
<tr id="S3.T2.17.15.17" class="ltx_tr">
<td id="S3.T2.17.15.17.1" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;" colspan="12">Proprietary Models</td>
</tr>
<tr id="S3.T2.17.15.18" class="ltx_tr">
<td id="S3.T2.17.15.18.1" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">GPT-4</td>
<td id="S3.T2.17.15.18.2" class="ltx_td ltx_align_right ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.18.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.18.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.18.4" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.18.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.18.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">92.0</td>
<td id="S3.T2.17.15.18.6" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">42.5</td>
<td id="S3.T2.17.15.18.7" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">64.7</td>
<td id="S3.T2.17.15.18.8" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">93.1</td>
<td id="S3.T2.17.15.18.9" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">67.1</td>
<td id="S3.T2.17.15.18.10" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">91.3</td>
<td id="S3.T2.17.15.18.11" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">97.6</td>
<td id="S3.T2.17.15.18.12" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">78.3</td>
</tr>
<tr id="S3.T2.4.2.2" class="ltx_tr">
<td id="S3.T2.4.2.2.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">GPT-4 (PAL) <span id="S3.T2.4.2.2.1.1" class="ltx_text" style="position:relative; bottom:-2.0pt;"><img src="/html/2309.17452/assets/figure/python.png" id="S3.T2.4.2.2.1.1.g1" class="ltx_graphics ltx_img_square" width="13" height="13" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T2.4.2.2.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.4.2.2.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.4.2.2.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.4.2.2.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.4.2.2.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.4.2.2.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">94.2</td>
<td id="S3.T2.4.2.2.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">51.8</td>
<td id="S3.T2.4.2.2.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">77.6</td>
<td id="S3.T2.4.2.2.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">94.8</td>
<td id="S3.T2.4.2.2.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">95.9</td>
<td id="S3.T2.4.2.2.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">92.6</td>
<td id="S3.T2.4.2.2.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">97.7</td>
<td id="S3.T2.4.2.2.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">86.4</td>
</tr>
<tr id="S3.T2.17.15.19" class="ltx_tr">
<td id="S3.T2.17.15.19.1" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">ChatGPT</td>
<td id="S3.T2.17.15.19.2" class="ltx_td ltx_align_right ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.19.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.19.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.19.4" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.19.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.19.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">80.8</td>
<td id="S3.T2.17.15.19.6" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">35.5</td>
<td id="S3.T2.17.15.19.7" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">55.9</td>
<td id="S3.T2.17.15.19.8" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">83.0</td>
<td id="S3.T2.17.15.19.9" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">69.1</td>
<td id="S3.T2.17.15.19.10" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">87.3</td>
<td id="S3.T2.17.15.19.11" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">94.6</td>
<td id="S3.T2.17.15.19.12" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">72.3</td>
</tr>
<tr id="S3.T2.5.3.3" class="ltx_tr">
<td id="S3.T2.5.3.3.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">ChatGPT (PAL) <span id="S3.T2.5.3.3.1.1" class="ltx_text" style="position:relative; bottom:-2.0pt;"><img src="/html/2309.17452/assets/figure/python.png" id="S3.T2.5.3.3.1.1.g1" class="ltx_graphics ltx_img_square" width="13" height="13" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T2.5.3.3.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.5.3.3.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.5.3.3.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.5.3.3.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.5.3.3.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.5.3.3.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">78.6</td>
<td id="S3.T2.5.3.3.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">38.7</td>
<td id="S3.T2.5.3.3.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">67.6</td>
<td id="S3.T2.5.3.3.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">77.8</td>
<td id="S3.T2.5.3.3.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">79.9</td>
<td id="S3.T2.5.3.3.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">81.0</td>
<td id="S3.T2.5.3.3.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">89.4</td>
<td id="S3.T2.5.3.3.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">73.3</td>
</tr>
<tr id="S3.T2.17.15.20" class="ltx_tr">
<td id="S3.T2.17.15.20.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Claude-2</td>
<td id="S3.T2.17.15.20.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.20.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.20.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.20.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.20.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.20.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">85.2</td>
<td id="S3.T2.17.15.20.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">32.5</td>
<td id="S3.T2.17.15.20.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.20.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.20.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.20.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.20.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.20.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
</tr>
<tr id="S3.T2.17.15.21" class="ltx_tr">
<td id="S3.T2.17.15.21.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">PaLM-2</td>
<td id="S3.T2.17.15.21.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">540B</td>
<td id="S3.T2.17.15.21.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.21.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.21.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.21.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.21.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">80.7</td>
<td id="S3.T2.17.15.21.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">34.3</td>
<td id="S3.T2.17.15.21.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.21.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.21.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.21.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.21.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.21.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
</tr>
<tr id="S3.T2.17.15.22" class="ltx_tr">
<td id="S3.T2.17.15.22.1" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;" colspan="12">Open-Source Models</td>
</tr>
<tr id="S3.T2.17.15.23" class="ltx_tr">
<td id="S3.T2.17.15.23.1" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">LLaMA-2</td>
<td id="S3.T2.17.15.23.2" class="ltx_td ltx_align_right ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">7B</td>
<td id="S3.T2.17.15.23.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.23.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.23.4" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.23.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.23.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">13.3</td>
<td id="S3.T2.17.15.23.6" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">4.1</td>
<td id="S3.T2.17.15.23.7" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">7.8</td>
<td id="S3.T2.17.15.23.8" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">38.0</td>
<td id="S3.T2.17.15.23.9" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">31.1</td>
<td id="S3.T2.17.15.23.10" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">50.7</td>
<td id="S3.T2.17.15.23.11" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">60.9</td>
<td id="S3.T2.17.15.23.12" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">29.4</td>
</tr>
<tr id="S3.T2.17.15.24" class="ltx_tr">
<td id="S3.T2.17.15.24.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">LLaMA-2 SFT</td>
<td id="S3.T2.17.15.24.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">7B</td>
<td id="S3.T2.17.15.24.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.24.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.24.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.24.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.17.15.24.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">41.3</td>
<td id="S3.T2.17.15.24.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">7.2</td>
<td id="S3.T2.17.15.24.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">16.1</td>
<td id="S3.T2.17.15.24.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">31.9</td>
<td id="S3.T2.17.15.24.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">27.8</td>
<td id="S3.T2.17.15.24.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">47.4</td>
<td id="S3.T2.17.15.24.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">60.0</td>
<td id="S3.T2.17.15.24.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">33.1</td>
</tr>
<tr id="S3.T2.17.15.25" class="ltx_tr">
<td id="S3.T2.17.15.25.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">LLaMA-2 RFT</td>
<td id="S3.T2.17.15.25.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">7B</td>
<td id="S3.T2.17.15.25.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.25.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.25.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.25.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.17.15.25.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">51.2</td>
<td id="S3.T2.17.15.25.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.25.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.25.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.25.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.25.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.25.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.25.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
</tr>
<tr id="S3.T2.17.15.26" class="ltx_tr">
<td id="S3.T2.17.15.26.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Platypus-2</td>
<td id="S3.T2.17.15.26.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">7B</td>
<td id="S3.T2.17.15.26.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.26.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.26.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.26.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.26.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">14.4</td>
<td id="S3.T2.17.15.26.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">5.4</td>
<td id="S3.T2.17.15.26.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">8.6</td>
<td id="S3.T2.17.15.26.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">36.7</td>
<td id="S3.T2.17.15.26.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">26.5</td>
<td id="S3.T2.17.15.26.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">47.9</td>
<td id="S3.T2.17.15.26.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">58.4</td>
<td id="S3.T2.17.15.26.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">28.3</td>
</tr>
<tr id="S3.T2.17.15.27" class="ltx_tr">
<td id="S3.T2.17.15.27.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">WizardMath</td>
<td id="S3.T2.17.15.27.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">7B</td>
<td id="S3.T2.17.15.27.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.27.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.27.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.27.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.17.15.27.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">54.9</td>
<td id="S3.T2.17.15.27.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">10.7</td>
<td id="S3.T2.17.15.27.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">20.6</td>
<td id="S3.T2.17.15.27.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">57.3</td>
<td id="S3.T2.17.15.27.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">38.1</td>
<td id="S3.T2.17.15.27.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">59.1</td>
<td id="S3.T2.17.15.27.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">73.7</td>
<td id="S3.T2.17.15.27.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">44.9</td>
</tr>
<tr id="S3.T2.6.4.4" class="ltx_tr">
<td id="S3.T2.6.4.4.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">CodeLLaMA (PAL) <span id="S3.T2.6.4.4.1.1" class="ltx_text" style="position:relative; bottom:-2.0pt;"><img src="/html/2309.17452/assets/figure/python.png" id="S3.T2.6.4.4.1.1.g1" class="ltx_graphics ltx_img_square" width="13" height="13" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T2.6.4.4.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">7B</td>
<td id="S3.T2.6.4.4.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.6.4.4.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.6.4.4.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.6.4.4.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.6.4.4.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">34.0</td>
<td id="S3.T2.6.4.4.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">16.6</td>
<td id="S3.T2.6.4.4.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">33.6</td>
<td id="S3.T2.6.4.4.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">59.0</td>
<td id="S3.T2.6.4.4.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.6.4.4.9.1" class="ltx_text ltx_framed ltx_framed_underline">47.3</span></td>
<td id="S3.T2.6.4.4.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">61.4</td>
<td id="S3.T2.6.4.4.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">79.6</td>
<td id="S3.T2.6.4.4.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">47.4</td>
</tr>
<tr id="S3.T2.8.6.6" class="ltx_tr">
<td id="S3.T2.8.6.6.2" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Toolformer<sup id="S3.T2.8.6.6.2.2" class="ltx_sup">†</sup> <span id="S3.T2.8.6.6.2.1" class="ltx_text" style="position:relative; bottom:-1.0pt;"><img src="/html/2309.17452/assets/figure/calculator.png" id="S3.T2.8.6.6.2.1.g1" class="ltx_graphics ltx_img_square" width="13" height="13" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T2.8.6.6.3" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">7B</td>
<td id="S3.T2.8.6.6.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.8.6.6.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.8.6.6.5" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.8.6.6.5.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.8.6.6.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.8.6.6.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.8.6.6.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.8.6.6.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">29.4</td>
<td id="S3.T2.8.6.6.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.8.6.6.11" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">40.4</td>
<td id="S3.T2.8.6.6.12" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">44.0</td>
<td id="S3.T2.8.6.6.13" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
</tr>
<tr id="S3.T2.9.7.7" class="ltx_tr">
<td id="S3.T2.9.7.7.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="S3.T2.9.7.7.1.2" class="ltx_text ltx_font_smallcaps">ToRA</span> <span id="S3.T2.9.7.7.1.1" class="ltx_text" style="position:relative; bottom:-4.3pt;"><img src="/html/2309.17452/assets/figure/tora_logo.png" id="S3.T2.9.7.7.1.1.g1" class="ltx_graphics ltx_img_portrait" width="16" height="20" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T2.9.7.7.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">7B</td>
<td id="S3.T2.9.7.7.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.9.7.7.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.9.7.7.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.9.7.7.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.9.7.7.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.9.7.7.5.1" class="ltx_text ltx_framed ltx_framed_underline">68.8</span></td>
<td id="S3.T2.9.7.7.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.9.7.7.6.1" class="ltx_text ltx_framed ltx_framed_underline">40.1</span></td>
<td id="S3.T2.9.7.7.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.9.7.7.7.1" class="ltx_text ltx_framed ltx_framed_underline">54.6</span></td>
<td id="S3.T2.9.7.7.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.9.7.7.8.1" class="ltx_text ltx_framed ltx_framed_underline">68.2</span></td>
<td id="S3.T2.9.7.7.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">42.4</td>
<td id="S3.T2.9.7.7.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.9.7.7.10.1" class="ltx_text ltx_framed ltx_framed_underline">73.9</span></td>
<td id="S3.T2.9.7.7.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.9.7.7.11.1" class="ltx_text ltx_framed ltx_framed_underline">88.8</span></td>
<td id="S3.T2.9.7.7.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.9.7.7.12.1" class="ltx_text ltx_framed ltx_framed_underline">62.4</span></td>
</tr>
<tr id="S3.T2.10.8.8" class="ltx_tr">
<td id="S3.T2.10.8.8.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="S3.T2.10.8.8.1.2" class="ltx_text ltx_font_smallcaps">ToRA-Code</span> <span id="S3.T2.10.8.8.1.1" class="ltx_text" style="position:relative; bottom:-4.3pt;"><img src="/html/2309.17452/assets/figure/tora_logo.png" id="S3.T2.10.8.8.1.1.g1" class="ltx_graphics ltx_img_portrait" width="16" height="20" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T2.10.8.8.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">7B</td>
<td id="S3.T2.10.8.8.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.10.8.8.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.10.8.8.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.10.8.8.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.10.8.8.5" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.10.8.8.5.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">72.6</span></td>
<td id="S3.T2.10.8.8.6" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.10.8.8.6.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">44.6</span></td>
<td id="S3.T2.10.8.8.7" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.10.8.8.7.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">56.0</span></td>
<td id="S3.T2.10.8.8.8" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.10.8.8.8.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">70.4</span></td>
<td id="S3.T2.10.8.8.9" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.10.8.8.9.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">51.6</span></td>
<td id="S3.T2.10.8.8.10" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.10.8.8.10.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">78.7</span></td>
<td id="S3.T2.10.8.8.11" class="ltx_td ltx_align_center ltx_border_r" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.10.8.8.11.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">91.3</span></td>
<td id="S3.T2.10.8.8.12" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.10.8.8.12.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">66.5 (+19)</span></td>
</tr>
<tr id="S3.T2.17.15.28" class="ltx_tr">
<td id="S3.T2.17.15.28.1" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">LLaMA-2</td>
<td id="S3.T2.17.15.28.2" class="ltx_td ltx_align_right ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">13B</td>
<td id="S3.T2.17.15.28.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.28.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.28.4" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.28.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.28.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">24.3</td>
<td id="S3.T2.17.15.28.6" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">6.3</td>
<td id="S3.T2.17.15.28.7" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">13.6</td>
<td id="S3.T2.17.15.28.8" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">43.1</td>
<td id="S3.T2.17.15.28.9" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">39.5</td>
<td id="S3.T2.17.15.28.10" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">56.3</td>
<td id="S3.T2.17.15.28.11" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">70.4</td>
<td id="S3.T2.17.15.28.12" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">36.2</td>
</tr>
<tr id="S3.T2.17.15.29" class="ltx_tr">
<td id="S3.T2.17.15.29.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">LLaMA-2 SFT</td>
<td id="S3.T2.17.15.29.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">13B</td>
<td id="S3.T2.17.15.29.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.29.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.29.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.29.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.17.15.29.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">51.1</td>
<td id="S3.T2.17.15.29.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">9.2</td>
<td id="S3.T2.17.15.29.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">22.3</td>
<td id="S3.T2.17.15.29.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">46.3</td>
<td id="S3.T2.17.15.29.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">35.8</td>
<td id="S3.T2.17.15.29.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">58.6</td>
<td id="S3.T2.17.15.29.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">75.0</td>
<td id="S3.T2.17.15.29.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">42.6</td>
</tr>
<tr id="S3.T2.17.15.30" class="ltx_tr">
<td id="S3.T2.17.15.30.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">LLaMA-2 RFT</td>
<td id="S3.T2.17.15.30.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">13B</td>
<td id="S3.T2.17.15.30.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.30.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.30.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.30.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.17.15.30.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">55.3</td>
<td id="S3.T2.17.15.30.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.30.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.30.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.30.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.30.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.30.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.30.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
</tr>
<tr id="S3.T2.17.15.31" class="ltx_tr">
<td id="S3.T2.17.15.31.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Platypus-2</td>
<td id="S3.T2.17.15.31.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">13B</td>
<td id="S3.T2.17.15.31.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.31.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.31.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.31.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.31.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">23.7</td>
<td id="S3.T2.17.15.31.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">7.1</td>
<td id="S3.T2.17.15.31.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">14.3</td>
<td id="S3.T2.17.15.31.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">50.7</td>
<td id="S3.T2.17.15.31.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">45.3</td>
<td id="S3.T2.17.15.31.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">55.1</td>
<td id="S3.T2.17.15.31.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">69.6</td>
<td id="S3.T2.17.15.31.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">38.0</td>
</tr>
<tr id="S3.T2.17.15.32" class="ltx_tr">
<td id="S3.T2.17.15.32.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">WizardMath</td>
<td id="S3.T2.17.15.32.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">13B</td>
<td id="S3.T2.17.15.32.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.32.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.32.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.32.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.17.15.32.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">63.9</td>
<td id="S3.T2.17.15.32.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">14.0</td>
<td id="S3.T2.17.15.32.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">28.4</td>
<td id="S3.T2.17.15.32.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">64.3</td>
<td id="S3.T2.17.15.32.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">46.7</td>
<td id="S3.T2.17.15.32.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">65.8</td>
<td id="S3.T2.17.15.32.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">79.7</td>
<td id="S3.T2.17.15.32.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">51.8</td>
</tr>
<tr id="S3.T2.11.9.9" class="ltx_tr">
<td id="S3.T2.11.9.9.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">CodeLLaMA (PAL) <span id="S3.T2.11.9.9.1.1" class="ltx_text" style="position:relative; bottom:-2.0pt;"><img src="/html/2309.17452/assets/figure/python.png" id="S3.T2.11.9.9.1.1.g1" class="ltx_graphics ltx_img_square" width="13" height="13" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T2.11.9.9.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">13B</td>
<td id="S3.T2.11.9.9.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.11.9.9.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.11.9.9.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.11.9.9.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.11.9.9.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">39.9</td>
<td id="S3.T2.11.9.9.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">19.9</td>
<td id="S3.T2.11.9.9.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">39.0</td>
<td id="S3.T2.11.9.9.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">62.4</td>
<td id="S3.T2.11.9.9.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.11.9.9.9.1" class="ltx_text ltx_framed ltx_framed_underline">59.5</span></td>
<td id="S3.T2.11.9.9.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">65.3</td>
<td id="S3.T2.11.9.9.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">86.0</td>
<td id="S3.T2.11.9.9.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">53.1</td>
</tr>
<tr id="S3.T2.12.10.10" class="ltx_tr">
<td id="S3.T2.12.10.10.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="S3.T2.12.10.10.1.2" class="ltx_text ltx_font_smallcaps">ToRA</span> <span id="S3.T2.12.10.10.1.1" class="ltx_text" style="position:relative; bottom:-4.3pt;"><img src="/html/2309.17452/assets/figure/tora_logo.png" id="S3.T2.12.10.10.1.1.g1" class="ltx_graphics ltx_img_portrait" width="16" height="20" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T2.12.10.10.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">13B</td>
<td id="S3.T2.12.10.10.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.12.10.10.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.12.10.10.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.12.10.10.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.12.10.10.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.12.10.10.5.1" class="ltx_text ltx_framed ltx_framed_underline">72.7</span></td>
<td id="S3.T2.12.10.10.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.12.10.10.6.1" class="ltx_text ltx_framed ltx_framed_underline">43.0</span></td>
<td id="S3.T2.12.10.10.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.12.10.10.7.1" class="ltx_text ltx_framed ltx_framed_underline">57.3</span></td>
<td id="S3.T2.12.10.10.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.12.10.10.8.1" class="ltx_text ltx_framed ltx_framed_underline">72.9</span></td>
<td id="S3.T2.12.10.10.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">47.2</td>
<td id="S3.T2.12.10.10.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.12.10.10.10.1" class="ltx_text ltx_framed ltx_framed_underline">77.2</span></td>
<td id="S3.T2.12.10.10.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.12.10.10.11.1" class="ltx_text ltx_framed ltx_framed_underline">91.3</span></td>
<td id="S3.T2.12.10.10.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.12.10.10.12.1" class="ltx_text ltx_framed ltx_framed_underline">65.9</span></td>
</tr>
<tr id="S3.T2.13.11.11" class="ltx_tr">
<td id="S3.T2.13.11.11.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="S3.T2.13.11.11.1.2" class="ltx_text ltx_font_smallcaps">ToRA-Code</span> <span id="S3.T2.13.11.11.1.1" class="ltx_text" style="position:relative; bottom:-4.3pt;"><img src="/html/2309.17452/assets/figure/tora_logo.png" id="S3.T2.13.11.11.1.1.g1" class="ltx_graphics ltx_img_portrait" width="16" height="20" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T2.13.11.11.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">13B</td>
<td id="S3.T2.13.11.11.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.13.11.11.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.13.11.11.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.13.11.11.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.13.11.11.5" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.13.11.11.5.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">75.8</span></td>
<td id="S3.T2.13.11.11.6" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.13.11.11.6.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">48.1</span></td>
<td id="S3.T2.13.11.11.7" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.13.11.11.7.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">60.5</span></td>
<td id="S3.T2.13.11.11.8" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.13.11.11.8.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">75.7</span></td>
<td id="S3.T2.13.11.11.9" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.13.11.11.9.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">65.4</span></td>
<td id="S3.T2.13.11.11.10" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.13.11.11.10.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">81.4</span></td>
<td id="S3.T2.13.11.11.11" class="ltx_td ltx_align_center ltx_border_r" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.13.11.11.11.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">92.5</span></td>
<td id="S3.T2.13.11.11.12" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.13.11.11.12.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">71.3 (+18)</span></td>
</tr>
<tr id="S3.T2.17.15.33" class="ltx_tr">
<td id="S3.T2.17.15.33.1" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">LLaMA-1 RFT</td>
<td id="S3.T2.17.15.33.2" class="ltx_td ltx_align_right ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">34B</td>
<td id="S3.T2.17.15.33.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.33.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.33.4" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.33.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.17.15.33.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">57.9</td>
<td id="S3.T2.17.15.33.6" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.33.7" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.33.8" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.33.9" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.33.10" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.33.11" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.33.12" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
</tr>
<tr id="S3.T2.14.12.12" class="ltx_tr">
<td id="S3.T2.14.12.12.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">CodeLLaMA (PAL) <span id="S3.T2.14.12.12.1.1" class="ltx_text" style="position:relative; bottom:-2.0pt;"><img src="/html/2309.17452/assets/figure/python.png" id="S3.T2.14.12.12.1.1.g1" class="ltx_graphics ltx_img_square" width="13" height="13" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T2.14.12.12.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">34B</td>
<td id="S3.T2.14.12.12.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.14.12.12.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.14.12.12.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.14.12.12.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.14.12.12.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">53.3</td>
<td id="S3.T2.14.12.12.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">23.9</td>
<td id="S3.T2.14.12.12.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">49.4</td>
<td id="S3.T2.14.12.12.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">71.0</td>
<td id="S3.T2.14.12.12.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">63.1</td>
<td id="S3.T2.14.12.12.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">72.4</td>
<td id="S3.T2.14.12.12.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">91.5</td>
<td id="S3.T2.14.12.12.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">60.7</td>
</tr>
<tr id="S3.T2.15.13.13" class="ltx_tr">
<td id="S3.T2.15.13.13.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="S3.T2.15.13.13.1.2" class="ltx_text ltx_font_smallcaps">ToRA-Code</span> <span id="S3.T2.15.13.13.1.1" class="ltx_text" style="position:relative; bottom:-4.3pt;"><img src="/html/2309.17452/assets/figure/tora_logo.png" id="S3.T2.15.13.13.1.1.g1" class="ltx_graphics ltx_img_portrait" width="16" height="20" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T2.15.13.13.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">34B</td>
<td id="S3.T2.15.13.13.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.15.13.13.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.15.13.13.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.15.13.13.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.15.13.13.5" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.15.13.13.5.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">80.7</span></td>
<td id="S3.T2.15.13.13.6" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.15.13.13.6.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">50.8</span></td>
<td id="S3.T2.15.13.13.7" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.15.13.13.7.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">63.7</span></td>
<td id="S3.T2.15.13.13.8" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.15.13.13.8.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">80.5</span></td>
<td id="S3.T2.15.13.13.9" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.15.13.13.9.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">70.5</span></td>
<td id="S3.T2.15.13.13.10" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.15.13.13.10.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">84.2</span></td>
<td id="S3.T2.15.13.13.11" class="ltx_td ltx_align_center ltx_border_r" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.15.13.13.11.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">93.3</span></td>
<td id="S3.T2.15.13.13.12" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.15.13.13.12.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">74.8 (+14)</span></td>
</tr>
<tr id="S3.T2.17.15.34" class="ltx_tr">
<td id="S3.T2.17.15.34.1" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">LLaMA-2</td>
<td id="S3.T2.17.15.34.2" class="ltx_td ltx_align_right ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">70B</td>
<td id="S3.T2.17.15.34.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.34.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.34.4" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.34.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.34.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">57.8</td>
<td id="S3.T2.17.15.34.6" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">14.4</td>
<td id="S3.T2.17.15.34.7" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">36.0</td>
<td id="S3.T2.17.15.34.8" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">73.6</td>
<td id="S3.T2.17.15.34.9" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">57.5</td>
<td id="S3.T2.17.15.34.10" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">76.0</td>
<td id="S3.T2.17.15.34.11" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">92.4</td>
<td id="S3.T2.17.15.34.12" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">58.2</td>
</tr>
<tr id="S3.T2.17.15.35" class="ltx_tr">
<td id="S3.T2.17.15.35.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">LLaMA-2 SFT</td>
<td id="S3.T2.17.15.35.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">70B</td>
<td id="S3.T2.17.15.35.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.35.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.35.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.35.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.17.15.35.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">69.3</td>
<td id="S3.T2.17.15.35.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">14.9</td>
<td id="S3.T2.17.15.35.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">39.0</td>
<td id="S3.T2.17.15.35.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">64.0</td>
<td id="S3.T2.17.15.35.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">53.0</td>
<td id="S3.T2.17.15.35.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">71.3</td>
<td id="S3.T2.17.15.35.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">84.8</td>
<td id="S3.T2.17.15.35.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">56.6</td>
</tr>
<tr id="S3.T2.17.15.36" class="ltx_tr">
<td id="S3.T2.17.15.36.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">LLaMA-2 RFT</td>
<td id="S3.T2.17.15.36.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">70B</td>
<td id="S3.T2.17.15.36.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.36.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.36.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.36.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.17.15.36.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">64.8</td>
<td id="S3.T2.17.15.36.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.36.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.36.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.36.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.36.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.36.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T2.17.15.36.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
</tr>
<tr id="S3.T2.17.15.37" class="ltx_tr">
<td id="S3.T2.17.15.37.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Platypus-2</td>
<td id="S3.T2.17.15.37.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">70B</td>
<td id="S3.T2.17.15.37.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.37.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.37.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.37.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.37.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">45.9</td>
<td id="S3.T2.17.15.37.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">15.0</td>
<td id="S3.T2.17.15.37.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">24.6</td>
<td id="S3.T2.17.15.37.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">74.3</td>
<td id="S3.T2.17.15.37.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">47.3</td>
<td id="S3.T2.17.15.37.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">72.7</td>
<td id="S3.T2.17.15.37.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">91.1</td>
<td id="S3.T2.17.15.37.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">53.0</td>
</tr>
<tr id="S3.T2.17.15.38" class="ltx_tr">
<td id="S3.T2.17.15.38.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">WizardMath</td>
<td id="S3.T2.17.15.38.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">70B</td>
<td id="S3.T2.17.15.38.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.38.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.17.15.38.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.38.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.17.15.38.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.38.5.1" class="ltx_text ltx_framed ltx_framed_underline">81.6</span></td>
<td id="S3.T2.17.15.38.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.38.6.1" class="ltx_text ltx_framed ltx_framed_underline">22.7</span></td>
<td id="S3.T2.17.15.38.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.38.7.1" class="ltx_text ltx_framed ltx_framed_underline">50.3</span></td>
<td id="S3.T2.17.15.38.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.38.8.1" class="ltx_text ltx_framed ltx_framed_underline">80.0</span></td>
<td id="S3.T2.17.15.38.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">49.8</td>
<td id="S3.T2.17.15.38.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.38.10.1" class="ltx_text ltx_framed ltx_framed_underline">76.2</span></td>
<td id="S3.T2.17.15.38.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">86.2</td>
<td id="S3.T2.17.15.38.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.38.12.1" class="ltx_text ltx_framed ltx_framed_underline">63.8</span></td>
</tr>
<tr id="S3.T2.16.14.14" class="ltx_tr">
<td id="S3.T2.16.14.14.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">LLaMA-2 (PAL) <span id="S3.T2.16.14.14.1.1" class="ltx_text" style="position:relative; bottom:-2.0pt;"><img src="/html/2309.17452/assets/figure/python.png" id="S3.T2.16.14.14.1.1.g1" class="ltx_graphics ltx_img_square" width="13" height="13" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T2.16.14.14.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">70B</td>
<td id="S3.T2.16.14.14.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.16.14.14.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.16.14.14.4" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.16.14.14.4.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T2.16.14.14.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">55.2</td>
<td id="S3.T2.16.14.14.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">18.3</td>
<td id="S3.T2.16.14.14.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">50.0</td>
<td id="S3.T2.16.14.14.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">74.6</td>
<td id="S3.T2.16.14.14.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.16.14.14.9.1" class="ltx_text ltx_framed ltx_framed_underline">59.5</span></td>
<td id="S3.T2.16.14.14.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">71.9</td>
<td id="S3.T2.16.14.14.11" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.16.14.14.11.1" class="ltx_text ltx_framed ltx_framed_underline">92.8</span></td>
<td id="S3.T2.16.14.14.12" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">60.3</td>
</tr>
<tr id="S3.T2.17.15.15" class="ltx_tr">
<td id="S3.T2.17.15.15.1" class="ltx_td ltx_align_left ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="S3.T2.17.15.15.1.2" class="ltx_text ltx_font_smallcaps">ToRA</span> <span id="S3.T2.17.15.15.1.1" class="ltx_text" style="position:relative; bottom:-4.3pt;"><img src="/html/2309.17452/assets/figure/tora_logo.png" id="S3.T2.17.15.15.1.1.g1" class="ltx_graphics ltx_img_portrait" width="16" height="20" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T2.17.15.15.2" class="ltx_td ltx_align_right ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">70B</td>
<td id="S3.T2.17.15.15.3" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.15.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.17.15.15.4" class="ltx_td ltx_align_center ltx_border_bb ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.15.4.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T2.17.15.15.5" class="ltx_td ltx_align_center ltx_border_bb" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.15.5.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">84.3</span></td>
<td id="S3.T2.17.15.15.6" class="ltx_td ltx_align_center ltx_border_bb" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.15.6.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">49.7</span></td>
<td id="S3.T2.17.15.15.7" class="ltx_td ltx_align_center ltx_border_bb" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.15.7.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">67.2</span></td>
<td id="S3.T2.17.15.15.8" class="ltx_td ltx_align_center ltx_border_bb" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.15.8.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">82.7</span></td>
<td id="S3.T2.17.15.15.9" class="ltx_td ltx_align_center ltx_border_bb" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.15.9.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">74.0</span></td>
<td id="S3.T2.17.15.15.10" class="ltx_td ltx_align_center ltx_border_bb" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.15.10.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">86.8</span></td>
<td id="S3.T2.17.15.15.11" class="ltx_td ltx_align_center ltx_border_bb ltx_border_r" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.15.11.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">93.8</span></td>
<td id="S3.T2.17.15.15.12" class="ltx_td ltx_align_center ltx_border_bb" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T2.17.15.15.12.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">76.9 (+13)</span></td>
</tr>
</table>
</span></div>
</figure>
</section>
<section id="S3.SS4" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">3.4 </span>Main Results</h3>

<figure id="S3.T3" class="ltx_table">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 3: </span>Results on MATH subtopics.
</figcaption>
<div id="S3.T3.6" class="ltx_inline-block ltx_align_center ltx_transformed_outer" style="width:433.6pt;height:251.3pt;vertical-align:-0.7pt;"><span class="ltx_transformed_inner" style="transform:translate(-114.9pt,66.4pt) scale(0.653642737091182,0.653642737091182) ;">
<table id="S3.T3.6.6" class="ltx_tabular ltx_align_middle">
<tr id="S3.T3.6.6.7" class="ltx_tr">
<td id="S3.T3.6.6.7.1" class="ltx_td ltx_align_left ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.7.1.1" class="ltx_text ltx_font_bold">Model</span></td>
<td id="S3.T3.6.6.7.2" class="ltx_td ltx_align_right ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.7.2.1" class="ltx_text ltx_font_bold">Size</span></td>
<td id="S3.T3.6.6.7.3" class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.7.3.1" class="ltx_text ltx_font_bold">Tool</span></td>
<td id="S3.T3.6.6.7.4" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;">
<table id="S3.T3.6.6.7.4.1" class="ltx_tabular ltx_align_middle">
<tr id="S3.T3.6.6.7.4.1.1" class="ltx_tr">
<td id="S3.T3.6.6.7.4.1.1.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.7.4.1.1.1.1" class="ltx_text ltx_font_bold">Intermediate</span></td>
</tr>
<tr id="S3.T3.6.6.7.4.1.2" class="ltx_tr">
<td id="S3.T3.6.6.7.4.1.2.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.7.4.1.2.1.1" class="ltx_text ltx_font_bold">Algebra</span></td>
</tr>
</table>
</td>
<td id="S3.T3.6.6.7.5" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.7.5.1" class="ltx_text ltx_font_bold">Precalculus</span></td>
<td id="S3.T3.6.6.7.6" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.7.6.1" class="ltx_text ltx_font_bold">Geometry</span></td>
<td id="S3.T3.6.6.7.7" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;">
<table id="S3.T3.6.6.7.7.1" class="ltx_tabular ltx_align_middle">
<tr id="S3.T3.6.6.7.7.1.1" class="ltx_tr">
<td id="S3.T3.6.6.7.7.1.1.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.7.7.1.1.1.1" class="ltx_text ltx_font_bold">Number</span></td>
</tr>
<tr id="S3.T3.6.6.7.7.1.2" class="ltx_tr">
<td id="S3.T3.6.6.7.7.1.2.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.7.7.1.2.1.1" class="ltx_text ltx_font_bold">Theory</span></td>
</tr>
</table>
</td>
<td id="S3.T3.6.6.7.8" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;">
<table id="S3.T3.6.6.7.8.1" class="ltx_tabular ltx_align_middle">
<tr id="S3.T3.6.6.7.8.1.1" class="ltx_tr">
<td id="S3.T3.6.6.7.8.1.1.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="S3.T3.6.6.7.8.1.1.1.1" class="ltx_text ltx_font_bold">Counting</span> &amp;</td>
</tr>
<tr id="S3.T3.6.6.7.8.1.2" class="ltx_tr">
<td id="S3.T3.6.6.7.8.1.2.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.7.8.1.2.1.1" class="ltx_text ltx_font_bold">Probability</span></td>
</tr>
</table>
</td>
<td id="S3.T3.6.6.7.9" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.7.9.1" class="ltx_text ltx_font_bold">Prealgebra</span></td>
<td id="S3.T3.6.6.7.10" class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.7.10.1" class="ltx_text ltx_font_bold">Algebra</span></td>
<td id="S3.T3.6.6.7.11" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.7.11.1" class="ltx_text ltx_font_bold">Overall</span></td>
</tr>
<tr id="S3.T3.6.6.8" class="ltx_tr">
<td id="S3.T3.6.6.8.1" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;" colspan="11">Proprietary Models</td>
</tr>
<tr id="S3.T3.1.1.1" class="ltx_tr">
<td id="S3.T3.1.1.1.1" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">ChatGPT (PAL) <span id="S3.T3.1.1.1.1.1" class="ltx_text" style="position:relative; bottom:-2.0pt;"><img src="/html/2309.17452/assets/figure/python.png" id="S3.T3.1.1.1.1.1.g1" class="ltx_graphics ltx_img_square" width="13" height="13" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T3.1.1.1.2" class="ltx_td ltx_align_right ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T3.1.1.1.3" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.1.1.1.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T3.1.1.1.4" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">18.5</td>
<td id="S3.T3.1.1.1.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">19.2</td>
<td id="S3.T3.1.1.1.6" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">23.2</td>
<td id="S3.T3.1.1.1.7" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">48.5</td>
<td id="S3.T3.1.1.1.8" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">43.0</td>
<td id="S3.T3.1.1.1.9" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">62.7</td>
<td id="S3.T3.1.1.1.10" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">45.4</td>
<td id="S3.T3.1.1.1.11" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">38.7</td>
</tr>
<tr id="S3.T3.2.2.2" class="ltx_tr">
<td id="S3.T3.2.2.2.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">GPT-4 (PAL) <span id="S3.T3.2.2.2.1.1" class="ltx_text" style="position:relative; bottom:-2.0pt;"><img src="/html/2309.17452/assets/figure/python.png" id="S3.T3.2.2.2.1.1.g1" class="ltx_graphics ltx_img_square" width="13" height="13" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T3.2.2.2.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">-</td>
<td id="S3.T3.2.2.2.3" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.2.2.2.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T3.2.2.2.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">32.8</td>
<td id="S3.T3.2.2.2.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">29.3</td>
<td id="S3.T3.2.2.2.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">38.0</td>
<td id="S3.T3.2.2.2.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">58.7</td>
<td id="S3.T3.2.2.2.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">61.0</td>
<td id="S3.T3.2.2.2.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">73.9</td>
<td id="S3.T3.2.2.2.10" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">59.1</td>
<td id="S3.T3.2.2.2.11" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">51.8</td>
</tr>
<tr id="S3.T3.6.6.9" class="ltx_tr">
<td id="S3.T3.6.6.9.1" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;" colspan="11">Open-Source Models</td>
</tr>
<tr id="S3.T3.6.6.10" class="ltx_tr">
<td id="S3.T3.6.6.10.1" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">WizardMath</td>
<td id="S3.T3.6.6.10.2" class="ltx_td ltx_align_right ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">7B</td>
<td id="S3.T3.6.6.10.3" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.10.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T3.6.6.10.4" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">6.2</td>
<td id="S3.T3.6.6.10.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">6.0</td>
<td id="S3.T3.6.6.10.6" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">6.5</td>
<td id="S3.T3.6.6.10.7" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">7.6</td>
<td id="S3.T3.6.6.10.8" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">9.5</td>
<td id="S3.T3.6.6.10.9" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">18.1</td>
<td id="S3.T3.6.6.10.10" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">16.3</td>
<td id="S3.T3.6.6.10.11" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">11.2</td>
</tr>
<tr id="S3.T3.3.3.3" class="ltx_tr">
<td id="S3.T3.3.3.3.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="S3.T3.3.3.3.1.2" class="ltx_text ltx_font_smallcaps">ToRA-Code</span> <span id="S3.T3.3.3.3.1.1" class="ltx_text" style="position:relative; bottom:-4.3pt;"><img src="/html/2309.17452/assets/figure/tora_logo.png" id="S3.T3.3.3.3.1.1.g1" class="ltx_graphics ltx_img_portrait" width="16" height="20" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T3.3.3.3.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">7B</td>
<td id="S3.T3.3.3.3.3" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.3.3.3.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T3.3.3.3.4" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.3.3.3.4.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">35.1 (+28.9)</span></td>
<td id="S3.T3.3.3.3.5" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.3.3.3.5.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">31.0 (+25.0)</span></td>
<td id="S3.T3.3.3.3.6" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.3.3.3.6.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">24.0 (+17.5)</span></td>
<td id="S3.T3.3.3.3.7" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.3.3.3.7.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">50.7 (+43.1)</span></td>
<td id="S3.T3.3.3.3.8" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.3.3.3.8.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">30.6 (+21.1)</span></td>
<td id="S3.T3.3.3.3.9" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.3.3.3.9.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">55.0 (+36.9)</span></td>
<td id="S3.T3.3.3.3.10" class="ltx_td ltx_align_center ltx_border_r" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.3.3.3.10.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">61.7 (+45.4)</span></td>
<td id="S3.T3.3.3.3.11" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.3.3.3.11.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">44.6 (+33.4)</span></td>
</tr>
<tr id="S3.T3.6.6.11" class="ltx_tr">
<td id="S3.T3.6.6.11.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">w/o Shaping</td>
<td id="S3.T3.6.6.11.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">7B</td>
<td id="S3.T3.6.6.11.3" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.11.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T3.6.6.11.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">29.7 (-5.4)</td>
<td id="S3.T3.6.6.11.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">25.1 (-5.9)</td>
<td id="S3.T3.6.6.11.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">17.7 (-6.3)</td>
<td id="S3.T3.6.6.11.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">46.9 (-3.8)</td>
<td id="S3.T3.6.6.11.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">32.3 (+1.7)</td>
<td id="S3.T3.6.6.11.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">51.9 (-3.1)</td>
<td id="S3.T3.6.6.11.10" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">55.7 (-6.0)</td>
<td id="S3.T3.6.6.11.11" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">40.2 (-4.4)</td>
</tr>
<tr id="S3.T3.6.6.12" class="ltx_tr">
<td id="S3.T3.6.6.12.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">w/o Rationale</td>
<td id="S3.T3.6.6.12.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">7B</td>
<td id="S3.T3.6.6.12.3" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.12.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T3.6.6.12.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">25.5 (-9.6)</td>
<td id="S3.T3.6.6.12.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">14.7 (-16.3)</td>
<td id="S3.T3.6.6.12.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">15.4 (-8.6)</td>
<td id="S3.T3.6.6.12.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">45.9 (-4.8)</td>
<td id="S3.T3.6.6.12.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">29.7 (-0.9)</td>
<td id="S3.T3.6.6.12.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">51.0 (-4.0)</td>
<td id="S3.T3.6.6.12.10" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">52.4 (-9.3)</td>
<td id="S3.T3.6.6.12.11" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">36.8 (-7.8)</td>
</tr>
<tr id="S3.T3.6.6.13" class="ltx_tr">
<td id="S3.T3.6.6.13.1" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">WizardMath</td>
<td id="S3.T3.6.6.13.2" class="ltx_td ltx_align_right ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">13B</td>
<td id="S3.T3.6.6.13.3" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.13.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T3.6.6.13.4" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">6.4</td>
<td id="S3.T3.6.6.13.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">6.6</td>
<td id="S3.T3.6.6.13.6" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">11.5</td>
<td id="S3.T3.6.6.13.7" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">9.6</td>
<td id="S3.T3.6.6.13.8" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">11.0</td>
<td id="S3.T3.6.6.13.9" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">28.5</td>
<td id="S3.T3.6.6.13.10" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">21.1</td>
<td id="S3.T3.6.6.13.11" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">15.0</td>
</tr>
<tr id="S3.T3.4.4.4" class="ltx_tr">
<td id="S3.T3.4.4.4.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="S3.T3.4.4.4.1.2" class="ltx_text ltx_font_smallcaps">ToRA-Code</span> <span id="S3.T3.4.4.4.1.1" class="ltx_text" style="position:relative; bottom:-4.3pt;"><img src="/html/2309.17452/assets/figure/tora_logo.png" id="S3.T3.4.4.4.1.1.g1" class="ltx_graphics ltx_img_portrait" width="16" height="20" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T3.4.4.4.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">13B</td>
<td id="S3.T3.4.4.4.3" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.4.4.4.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T3.4.4.4.4" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.4.4.4.4.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">35.7 (+29.3)</span></td>
<td id="S3.T3.4.4.4.5" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.4.4.4.5.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">31.1 (+24.5)</span></td>
<td id="S3.T3.4.4.4.6" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.4.4.4.6.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">25.7 (+14.2)</span></td>
<td id="S3.T3.4.4.4.7" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.4.4.4.7.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">55.6 (+46.0)</span></td>
<td id="S3.T3.4.4.4.8" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.4.4.4.8.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">39.5 (+28.5)</span></td>
<td id="S3.T3.4.4.4.9" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.4.4.4.9.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">58.7 (+30.2)</span></td>
<td id="S3.T3.4.4.4.10" class="ltx_td ltx_align_center ltx_border_r" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.4.4.4.10.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">66.7 (+45.6)</span></td>
<td id="S3.T3.4.4.4.11" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.4.4.4.11.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">48.1 (+33.1)</span></td>
</tr>
<tr id="S3.T3.6.6.14" class="ltx_tr">
<td id="S3.T3.6.6.14.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">w/o Shaping</td>
<td id="S3.T3.6.6.14.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">13B</td>
<td id="S3.T3.6.6.14.3" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.14.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T3.6.6.14.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">32.8 (-2.9)</td>
<td id="S3.T3.6.6.14.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">26.0 (-5.1)</td>
<td id="S3.T3.6.6.14.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">24.0 (-1.7)</td>
<td id="S3.T3.6.6.14.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">52.6 (-3.0)</td>
<td id="S3.T3.6.6.14.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">38.4 (-1.1)</td>
<td id="S3.T3.6.6.14.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">55.6 (-3.1)</td>
<td id="S3.T3.6.6.14.10" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">61.2 (-5.5)</td>
<td id="S3.T3.6.6.14.11" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">44.6 (-3.5)</td>
</tr>
<tr id="S3.T3.6.6.15" class="ltx_tr">
<td id="S3.T3.6.6.15.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">w/o Rationale</td>
<td id="S3.T3.6.6.15.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">13B</td>
<td id="S3.T3.6.6.15.3" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.15.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T3.6.6.15.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">27.1 (-8.6)</td>
<td id="S3.T3.6.6.15.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">15.8 (-15.3)</td>
<td id="S3.T3.6.6.15.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">16.3 (-9.4)</td>
<td id="S3.T3.6.6.15.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">50.4 (-5.2)</td>
<td id="S3.T3.6.6.15.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">36.9 (-2.6)</td>
<td id="S3.T3.6.6.15.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">55.3 (-3.4)</td>
<td id="S3.T3.6.6.15.10" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">56.5 (-10.2)</td>
<td id="S3.T3.6.6.15.11" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">40.2 (-7.9)</td>
</tr>
<tr id="S3.T3.5.5.5" class="ltx_tr">
<td id="S3.T3.5.5.5.1" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="S3.T3.5.5.5.1.2" class="ltx_text ltx_font_smallcaps">ToRA-Code</span> <span id="S3.T3.5.5.5.1.1" class="ltx_text" style="position:relative; bottom:-4.3pt;"><img src="/html/2309.17452/assets/figure/tora_logo.png" id="S3.T3.5.5.5.1.1.g1" class="ltx_graphics ltx_img_portrait" width="16" height="20" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T3.5.5.5.2" class="ltx_td ltx_align_right ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">34B</td>
<td id="S3.T3.5.5.5.3" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.5.5.5.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T3.5.5.5.4" class="ltx_td ltx_align_center ltx_border_t" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.5.5.5.4.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">38.9</span></td>
<td id="S3.T3.5.5.5.5" class="ltx_td ltx_align_center ltx_border_t" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.5.5.5.5.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">34.6</span></td>
<td id="S3.T3.5.5.5.6" class="ltx_td ltx_align_center ltx_border_t" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.5.5.5.6.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">27.3</span></td>
<td id="S3.T3.5.5.5.7" class="ltx_td ltx_align_center ltx_border_t" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.5.5.5.7.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">57.8</span></td>
<td id="S3.T3.5.5.5.8" class="ltx_td ltx_align_center ltx_border_t" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.5.5.5.8.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">41.4</span></td>
<td id="S3.T3.5.5.5.9" class="ltx_td ltx_align_center ltx_border_t" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.5.5.5.9.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">63.7</span></td>
<td id="S3.T3.5.5.5.10" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.5.5.5.10.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">67.7</span></td>
<td id="S3.T3.5.5.5.11" class="ltx_td ltx_align_center ltx_border_t" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.5.5.5.11.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">50.8</span></td>
</tr>
<tr id="S3.T3.6.6.16" class="ltx_tr">
<td id="S3.T3.6.6.16.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">w/o Shaping</td>
<td id="S3.T3.6.6.16.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">34B</td>
<td id="S3.T3.6.6.16.3" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.16.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T3.6.6.16.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">34.0 (-4.9)</td>
<td id="S3.T3.6.6.16.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">29.9 (-4.7)</td>
<td id="S3.T3.6.6.16.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">24.6 (-2.7)</td>
<td id="S3.T3.6.6.16.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">55.6 (-2.2)</td>
<td id="S3.T3.6.6.16.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">41.6 (+0.2)</td>
<td id="S3.T3.6.6.16.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">63.8 (+0.1)</td>
<td id="S3.T3.6.6.16.10" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">61.4 (-6.3)</td>
<td id="S3.T3.6.6.16.11" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">47.4 (-3.4)</td>
</tr>
<tr id="S3.T3.6.6.17" class="ltx_tr">
<td id="S3.T3.6.6.17.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">w/o Rationale</td>
<td id="S3.T3.6.6.17.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">34B</td>
<td id="S3.T3.6.6.17.3" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.17.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T3.6.6.17.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">28.3 (-10.6)</td>
<td id="S3.T3.6.6.17.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">15.8 (-18.8)</td>
<td id="S3.T3.6.6.17.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">18.0 (-9.3)</td>
<td id="S3.T3.6.6.17.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">52.4 (-5.4)</td>
<td id="S3.T3.6.6.17.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">40.7 (-0.7)</td>
<td id="S3.T3.6.6.17.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">58.6 (-5.1)</td>
<td id="S3.T3.6.6.17.10" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">57.5 (-10.2)</td>
<td id="S3.T3.6.6.17.11" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">41.9 (-8.9)</td>
</tr>
<tr id="S3.T3.6.6.18" class="ltx_tr">
<td id="S3.T3.6.6.18.1" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">WizardMath</td>
<td id="S3.T3.6.6.18.2" class="ltx_td ltx_align_right ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">70B</td>
<td id="S3.T3.6.6.18.3" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.18.3.1" class="ltx_text" style="color:#C80000;">✗</span></td>
<td id="S3.T3.6.6.18.4" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">9.1</td>
<td id="S3.T3.6.6.18.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">13.4</td>
<td id="S3.T3.6.6.18.6" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">16.9</td>
<td id="S3.T3.6.6.18.7" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">16.5</td>
<td id="S3.T3.6.6.18.8" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">19.2</td>
<td id="S3.T3.6.6.18.9" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">42.7</td>
<td id="S3.T3.6.6.18.10" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">35.0</td>
<td id="S3.T3.6.6.18.11" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">24.1</td>
</tr>
<tr id="S3.T3.6.6.6" class="ltx_tr">
<td id="S3.T3.6.6.6.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="S3.T3.6.6.6.1.2" class="ltx_text ltx_font_smallcaps">ToRA</span> <span id="S3.T3.6.6.6.1.1" class="ltx_text" style="position:relative; bottom:-4.3pt;"><img src="/html/2309.17452/assets/figure/tora_logo.png" id="S3.T3.6.6.6.1.1.g1" class="ltx_graphics ltx_img_portrait" width="16" height="20" alt="[Uncaptioned image]"></span>
</td>
<td id="S3.T3.6.6.6.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">70B</td>
<td id="S3.T3.6.6.6.3" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.6.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T3.6.6.6.4" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.6.4.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">37.1 (+28)</span></td>
<td id="S3.T3.6.6.6.5" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.6.5.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">30.4 (+17)</span></td>
<td id="S3.T3.6.6.6.6" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.6.6.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">30.1 (+13.2)</span></td>
<td id="S3.T3.6.6.6.7" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.6.7.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">54.6 (+38.1)</span></td>
<td id="S3.T3.6.6.6.8" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.6.8.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">40.3 (+21.1)</span></td>
<td id="S3.T3.6.6.6.9" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.6.9.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">64.9 (+22.2)</span></td>
<td id="S3.T3.6.6.6.10" class="ltx_td ltx_align_center ltx_border_r" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.6.10.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">66.6 (+31.6)</span></td>
<td id="S3.T3.6.6.6.11" class="ltx_td ltx_align_center" style="background-color:#D2DCFA;padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.6.11.1" class="ltx_text ltx_font_bold" style="background-color:#D2DCFA;">49.7 (+25.6)</span></td>
</tr>
<tr id="S3.T3.6.6.19" class="ltx_tr">
<td id="S3.T3.6.6.19.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">w/o Shaping</td>
<td id="S3.T3.6.6.19.2" class="ltx_td ltx_align_right" style="padding-left:4.0pt;padding-right:4.0pt;">70B</td>
<td id="S3.T3.6.6.19.3" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.19.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T3.6.6.19.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">33.8(-3.3)</td>
<td id="S3.T3.6.6.19.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">28.9(-1.5)</td>
<td id="S3.T3.6.6.19.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">27.1(-3)</td>
<td id="S3.T3.6.6.19.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">53.0(-1.6)</td>
<td id="S3.T3.6.6.19.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">38.0(-2.3)</td>
<td id="S3.T3.6.6.19.9" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">62.2(-2.7)</td>
<td id="S3.T3.6.6.19.10" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">64.2(-2.4)</td>
<td id="S3.T3.6.6.19.11" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">47.3(-2.4)</td>
</tr>
<tr id="S3.T3.6.6.20" class="ltx_tr">
<td id="S3.T3.6.6.20.1" class="ltx_td ltx_align_left ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">w/o Rationale</td>
<td id="S3.T3.6.6.20.2" class="ltx_td ltx_align_right ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">70B</td>
<td id="S3.T3.6.6.20.3" class="ltx_td ltx_align_center ltx_border_bb ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T3.6.6.20.3.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="S3.T3.6.6.20.4" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">26.7(-10.4)</td>
<td id="S3.T3.6.6.20.5" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">14.7(-15.7)</td>
<td id="S3.T3.6.6.20.6" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">20.3(-9.8)</td>
<td id="S3.T3.6.6.20.7" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">48.9(-5.7)</td>
<td id="S3.T3.6.6.20.8" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">39.2(-1.1)</td>
<td id="S3.T3.6.6.20.9" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">59.8(-5.1)</td>
<td id="S3.T3.6.6.20.10" class="ltx_td ltx_align_center ltx_border_bb ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">57.6(-9)</td>
<td id="S3.T3.6.6.20.11" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">41.5(-8.2)</td>
</tr>
</table>
</span></div>
</figure>
<div id="S3.SS4.p1" class="ltx_para ltx_noindent">
<p id="S3.SS4.p1.1" class="ltx_p">Table <a href="#S3.T2" title="Table 2 ‣ 3.3 Baselines ‣ 3 Experiments ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a> presents the results of <span id="S3.SS4.p1.1.1" class="ltx_text ltx_font_smallcaps">ToRA</span> on 10 mathematical datasets, highlighting the following salient observations:
<span id="S3.SS4.p1.1.2" class="ltx_text ltx_font_bold">(1)</span> Using interleaved formatting and output space shaping, <span id="S3.SS4.p1.1.3" class="ltx_text ltx_font_smallcaps">ToRA</span> consistently surpasses prior state-of-the-art open-source models across all scales, achieving 13% to 19% absolute improvements across 10 tasks.
<span id="S3.SS4.p1.1.4" class="ltx_text ltx_font_bold">(2)</span> <span id="S3.SS4.p1.1.5" class="ltx_text ltx_font_smallcaps">ToRA</span>-70B substantially outperforms ChatGPT with both CoT and PAL prompting on GSM8k (84.3% vs. 80.4%) and MATH (49.7% vs. 38.7%), while <span id="S3.SS4.p1.1.6" class="ltx_text ltx_font_smallcaps">ToRA-Code</span>-34B is competitive with GPT-4 solving competition-level MATH dataset with code (50.8% vs. 51.8%).
<span id="S3.SS4.p1.1.7" class="ltx_text ltx_font_bold">(3)</span> The accuracy of <span id="S3.SS4.p1.1.8" class="ltx_text ltx_font_smallcaps">ToRA-Code</span> is about 5% higher than <span id="S3.SS4.p1.1.9" class="ltx_text ltx_font_smallcaps">ToRA</span> of the same size, demonstrating that continued training on code data significantly benefits program-based tool use.
<span id="S3.SS4.p1.1.10" class="ltx_text ltx_font_bold">(4)</span> While rationale-based fine-tuning negatively affects out-of-distribution generalization, <span id="S3.SS4.p1.1.11" class="ltx_text ltx_font_smallcaps">ToRA</span> displays superior generalization. For instance, WizardMath-70B underperforms the base model on TabMWP (49.8% vs. 57.5%), while <span id="S3.SS4.p1.1.12" class="ltx_text ltx_font_smallcaps">ToRA</span>-70B effectively generalizes to this tabular reasoning task (74.0%).
<span id="S3.SS4.p1.1.13" class="ltx_text ltx_font_bold">(5)</span> <span id="S3.SS4.p1.1.14" class="ltx_text ltx_font_smallcaps">ToRA</span> attains fast zero-shot inference speed, averaging 1.02 tool interaction rounds per problem, while effectively addressing problems that require interactive tool utilization.</p>
</div>
</section>
<section id="S3.SS5" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">3.5 </span>Ablation Study</h3>

<figure id="S3.F4" class="ltx_figure"><img src="/html/2309.17452/assets/x4.png" id="S3.F4.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="422" height="140" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>
Comparison of three formats:
(1) Rationale-only: step-by-step natural language reasoning like CoT;
(2) Program-only: solving problems with programs like PAL;
(3) Tool-integrated Reasoning used by <span id="S3.F4.3.1" class="ltx_text ltx_font_smallcaps">ToRA</span>: interweaving rationale and program execution to solve problems.
We evaluated GPT-4 with few-shot prompting.
We trained LLaMA-2 models to reason in the three types of formats, respectively.
For a fair comparison, we <em id="S3.F4.4.2" class="ltx_emph ltx_font_italic">do not apply output space shaping</em> for all LLaMA-2 models.</figcaption>
</figure>
<section id="S3.SS5.SSS1" class="ltx_subsubsection">
<h4 class="ltx_title ltx_title_subsubsection">
<span class="ltx_tag ltx_tag_subsubsection">3.5.1 </span>Comparisons of Formatting</h4>

<div id="S3.SS5.SSS1.p1" class="ltx_para ltx_noindent">
<p id="S3.SS5.SSS1.p1.1" class="ltx_p">To evaluate the efficacy of the reasoning format adopted by <span id="S3.SS5.SSS1.p1.1.1" class="ltx_text ltx_font_smallcaps">ToRA</span> which interleaves rationales with programs, we compared it with Rationale-only and Program-only formats using GPT-4 and LLaMA-2 trained with the same size of data from MATH.
As shown in Fig <a href="#S3.F4" title="Figure 4 ‣ 3.5 Ablation Study ‣ 3 Experiments ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4</span></a>, the <span id="S3.SS5.SSS1.p1.1.2" class="ltx_text ltx_font_smallcaps">ToRA</span> method consistently surpasses Rationale-only and Program-only approaches.
Remarkably, using LLaMA-2, the <span id="S3.SS5.SSS1.p1.1.3" class="ltx_text ltx_font_smallcaps">ToRA</span> method achieves substantial improvements of 29.0% and 6.7% over Rationale-only and Program-only, respectively.
With the closed-source GPT-4, the improvements are 19.1% and 9.8%, respectively.
This emphasizes the effectiveness of integrating natural language rationales with programs.</p>
</div>
</section>
<section id="S3.SS5.SSS2" class="ltx_subsubsection">
<h4 class="ltx_title ltx_title_subsubsection">
<span class="ltx_tag ltx_tag_subsubsection">3.5.2 </span>Effects of Output Space Shaping</h4>

<figure id="S3.F5" class="ltx_figure"><img src="/html/2309.17452/assets/x5.png" id="S3.F5.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="422" height="113" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 5: </span>Ablation on output space shaping strategies using CodeLLaMA:
(1) <span id="S3.F5.3.1" class="ltx_text ltx_font_smallcaps">ToRA<math id="S3.F5.3.1.m1.1" class="ltx_Math" alttext="{}_{-\text{Correction}}^{-\text{Sampling}}" display="inline"><semantics id="S3.F5.3.1.m1.1b"><mmultiscripts id="S3.F5.3.1.m1.1.1" xref="S3.F5.3.1.m1.1.1.cmml"><mi id="S3.F5.3.1.m1.1.1.2.2" xref="S3.F5.3.1.m1.1.1.2.2.cmml"></mi><mprescripts id="S3.F5.3.1.m1.1.1b" xref="S3.F5.3.1.m1.1.1.cmml"></mprescripts><mrow id="S3.F5.3.1.m1.1.1c" xref="S3.F5.3.1.m1.1.1.cmml"></mrow><mrow id="S3.F5.3.1.m1.1.1.3" xref="S3.F5.3.1.m1.1.1.3.cmml"><mo id="S3.F5.3.1.m1.1.1.3b" xref="S3.F5.3.1.m1.1.1.3.cmml">−</mo><mtext id="S3.F5.3.1.m1.1.1.3.2" xref="S3.F5.3.1.m1.1.1.3.2a.cmml">Sampling</mtext></mrow><mrow id="S3.F5.3.1.m1.1.1.2.3" xref="S3.F5.3.1.m1.1.1.2.3.cmml"><mo id="S3.F5.3.1.m1.1.1.2.3b" xref="S3.F5.3.1.m1.1.1.2.3.cmml">−</mo><mtext id="S3.F5.3.1.m1.1.1.2.3.2" xref="S3.F5.3.1.m1.1.1.2.3.2a.cmml">Correction</mtext></mrow><mrow id="S3.F5.3.1.m1.1.1d" xref="S3.F5.3.1.m1.1.1.cmml"></mrow></mmultiscripts><annotation-xml encoding="MathML-Content" id="S3.F5.3.1.m1.1c"><apply id="S3.F5.3.1.m1.1.1.cmml" xref="S3.F5.3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.F5.3.1.m1.1.1.1.cmml" xref="S3.F5.3.1.m1.1.1">superscript</csymbol><apply id="S3.F5.3.1.m1.1.1.2.cmml" xref="S3.F5.3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.F5.3.1.m1.1.1.2.1.cmml" xref="S3.F5.3.1.m1.1.1">subscript</csymbol><csymbol cd="latexml" id="S3.F5.3.1.m1.1.1.2.2.cmml" xref="S3.F5.3.1.m1.1.1.2.2">absent</csymbol><apply id="S3.F5.3.1.m1.1.1.2.3.cmml" xref="S3.F5.3.1.m1.1.1.2.3"><minus id="S3.F5.3.1.m1.1.1.2.3.1.cmml" xref="S3.F5.3.1.m1.1.1.2.3"></minus><ci id="S3.F5.3.1.m1.1.1.2.3.2a.cmml" xref="S3.F5.3.1.m1.1.1.2.3.2"><mtext mathsize="70%" id="S3.F5.3.1.m1.1.1.2.3.2.cmml" xref="S3.F5.3.1.m1.1.1.2.3.2">Correction</mtext></ci></apply></apply><apply id="S3.F5.3.1.m1.1.1.3.cmml" xref="S3.F5.3.1.m1.1.1.3"><minus id="S3.F5.3.1.m1.1.1.3.1.cmml" xref="S3.F5.3.1.m1.1.1.3"></minus><ci id="S3.F5.3.1.m1.1.1.3.2a.cmml" xref="S3.F5.3.1.m1.1.1.3.2"><mtext mathsize="70%" id="S3.F5.3.1.m1.1.1.3.2.cmml" xref="S3.F5.3.1.m1.1.1.3.2">Sampling</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.3.1.m1.1d">{}_{-\text{Correction}}^{-\text{Sampling}}</annotation></semantics></math></span> is trained on <span id="S3.F5.7.3" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span> without shaping. (2) <span id="S3.F5.4.2" class="ltx_text ltx_font_smallcaps">ToRA<math id="S3.F5.4.2.m1.1" class="ltx_Math" alttext="{}_{-\text{Correction}}" display="inline"><semantics id="S3.F5.4.2.m1.1b"><msub id="S3.F5.4.2.m1.1.1" xref="S3.F5.4.2.m1.1.1.cmml"><mi id="S3.F5.4.2.m1.1.1b" xref="S3.F5.4.2.m1.1.1.cmml"></mi><mrow id="S3.F5.4.2.m1.1.1.1" xref="S3.F5.4.2.m1.1.1.1.cmml"><mo id="S3.F5.4.2.m1.1.1.1b" xref="S3.F5.4.2.m1.1.1.1.cmml">−</mo><mtext id="S3.F5.4.2.m1.1.1.1.2" xref="S3.F5.4.2.m1.1.1.1.2a.cmml">Correction</mtext></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.F5.4.2.m1.1c"><apply id="S3.F5.4.2.m1.1.1.cmml" xref="S3.F5.4.2.m1.1.1"><apply id="S3.F5.4.2.m1.1.1.1.cmml" xref="S3.F5.4.2.m1.1.1.1"><minus id="S3.F5.4.2.m1.1.1.1.1.cmml" xref="S3.F5.4.2.m1.1.1.1"></minus><ci id="S3.F5.4.2.m1.1.1.1.2a.cmml" xref="S3.F5.4.2.m1.1.1.1.2"><mtext mathsize="70%" id="S3.F5.4.2.m1.1.1.1.2.cmml" xref="S3.F5.4.2.m1.1.1.1.2">Correction</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.4.2.m1.1d">{}_{-\text{Correction}}</annotation></semantics></math></span> employs only the sampling strategy for shaping, trained with up to 4 additional valid trajectory samples per problem. (3) <span id="S3.F5.8.4" class="ltx_text ltx_font_smallcaps">ToRA</span> utilizes both the sampling and correction, also trained with up to 4 additional trajectories per problem.</figcaption>
</figure>
<div id="S3.SS5.SSS2.p1" class="ltx_para ltx_noindent">
<p id="S3.SS5.SSS2.p1.1" class="ltx_p">We assess the effectiveness of the output space shaping strategies presented in Section <a href="#S2.SS3.SSS0.Px2" title="Output Space Shaping ‣ 2.3 Training ‣ 2 ToRA: Tool-Integrated Agents for Mathematical Reasoning ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2.3</span></a>, specifically sampling and correction.
As shown in Fig <a href="#S3.F5" title="Figure 5 ‣ 3.5.2 Effects of Output Space Shaping ‣ 3.5 Ablation Study ‣ 3 Experiments ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">5</span></a> and Table <a href="#S3.T3" title="Table 3 ‣ 3.4 Main Results ‣ 3 Experiments ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3</span></a>:
(1) Output space shaping yields a considerable average improvement of 3.4% and 4.0% absolute for GSM8k and MATH, respectively, with greater benefits for smaller models;
(2) Applying the sampling strategy results in a 2.7% absolute improvement on average, while additionally incorporating correction offers a more substantial boost of up to 4.5%, without using more training data;
(3) Output space shaping benefits even the largest model <span id="S3.SS5.SSS2.p1.1.1" class="ltx_text ltx_font_smallcaps">ToRA</span>-70B, with a notable improvement from 47.3% to 49.7% on MATH.
These findings highlight the effectiveness of our shaping strategies across different model sizes and datasets.</p>
</div>
</section>
</section>
<section id="S3.SS6" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">3.6 </span>Analysis</h3>

<div id="S3.SS6.p1" class="ltx_para ltx_noindent">
<p id="S3.SS6.p1.1" class="ltx_p">We investigate the benefits, detailed patterns, and remaining challenges of tool interaction for mathematical reasoning on the challenging MATH dataset.
Performance breakdowns on all subtopics of MATH are reported in Table <a href="#S3.T3" title="Table 3 ‣ 3.4 Main Results ‣ 3 Experiments ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3</span></a>.</p>
</div>
<div id="S3.SS6.p2" class="ltx_para ltx_noindent">
<p id="S3.SS6.p2.1" class="ltx_p"><span id="S3.SS6.p2.1.1" class="ltx_text ltx_font_bold">Benefits from Tool-Integration for MATH Sub-topics</span>
As shown in Table <a href="#S3.T3" title="Table 3 ‣ 3.4 Main Results ‣ 3 Experiments ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3</span></a>, <span id="S3.SS6.p2.1.2" class="ltx_text ltx_font_smallcaps">ToRA</span> outperforms WizardMath by around 45% in Algebra and Number Theory, which is attributed to stimulating and shaping tool-use behavior.
Problems from the two sub-topics typically need intricate computation and data manipulation.
Algebra mainly focuses on solving equations and application problems, while many Number Theory problems can be tackled using brute-force approaches through code.</p>
</div>
<figure id="S3.F6" class="ltx_figure"><img src="/html/2309.17452/assets/x6.png" id="S3.F6.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="422" height="137" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 6: </span>Library usage frequency and accuracy on each sub-topic of MATH.</figcaption>
</figure>
<div id="S3.SS6.p3" class="ltx_para ltx_noindent">
<p id="S3.SS6.p3.1" class="ltx_p"><span id="S3.SS6.p3.1.1" class="ltx_text ltx_font_bold">Patterns of Library Usage for Problem Solving</span>
Fig <a href="#S3.F6" title="Figure 6 ‣ 3.6 Analysis ‣ 3 Experiments ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">6</span></a> presents the most frequently used libraries for different sub-topics and the corresponding accuracies of their solutions.
Tool-use behavior on different mathematical areas demonstrates distinct patterns.
<span id="S3.SS6.p3.1.2" class="ltx_text ltx_font_typewriter">sympy</span> and its internal <span id="S3.SS6.p3.1.3" class="ltx_text ltx_font_typewriter">solvers</span> are primarily employed for algebra-related topics.
Precalculus exhibits extensive matrix operations via <span id="S3.SS6.p3.1.4" class="ltx_text ltx_font_typewriter">matrices</span>, resulting in a high accuracy.
Number Theory depends on <span id="S3.SS6.p3.1.5" class="ltx_text ltx_font_typewriter">algorithms</span> like <span id="S3.SS6.p3.1.6" class="ltx_text ltx_font_typewriter">gcd</span> and <span id="S3.SS6.p3.1.7" class="ltx_text ltx_font_typewriter">lcm</span>.
Geometry mainly uses the <span id="S3.SS6.p3.1.8" class="ltx_text ltx_font_typewriter">rational</span> library for fraction-based computations, while the application of other tools is limited, signifying the potential for improvement.</p>
</div>
<div id="S3.SS6.p4" class="ltx_para ltx_noindent">
<p id="S3.SS6.p4.1" class="ltx_p"><span id="S3.SS6.p4.1.1" class="ltx_text ltx_font_bold">Detailed Impact of Rationale on Different Topics</span>
Table <a href="#S3.T3" title="Table 3 ‣ 3.4 Main Results ‣ 3 Experiments ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3</span></a> shows that using an interleaved format, in contrast to merely writing the program, leads to significant improvements across all subtopics, especially in Precalculus, Algebra, and Geometry, where notable increases range from 8.6% to 18.8%. Appendix <a href="#A6.SS1" title="F.1 Success Cases ‣ Appendix F Examples ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">F.1</span></a> provides representative examples demonstrating how the rationale aids in planning, multi-round self-correction, and finalizing answers.</p>
</div>
<figure id="S3.T4" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 4: </span>
The failure modes of the <span id="S3.T4.2.1" class="ltx_text ltx_font_smallcaps">ToRA</span> on MATH, and their corresponding percentages in random samples analyzed by humans. We include specific examples of each failure mode in Appendix <a href="#A6" title="Appendix F Examples ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">F</span></a>.
</figcaption>
<div id="S3.T4.3" class="ltx_inline-block ltx_align_center ltx_transformed_outer" style="width:433.6pt;height:167.1pt;vertical-align:-0.9pt;"><span class="ltx_transformed_inner" style="transform:translate(-32.1pt,12.3pt) scale(0.871109900219567,0.871109900219567) ;">
<table id="S3.T4.3.1" class="ltx_tabular ltx_align_middle">
<tr id="S3.T4.3.1.1" class="ltx_tr">
<td id="S3.T4.3.1.1.1" class="ltx_td ltx_align_left ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T4.3.1.1.1.1" class="ltx_text ltx_font_bold">Error Type</span></td>
<td id="S3.T4.3.1.1.2" class="ltx_td ltx_align_left ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T4.3.1.1.2.1" class="ltx_text ltx_font_bold">Definition</span></td>
<td id="S3.T4.3.1.1.3" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="S3.T4.3.1.1.3.1" class="ltx_text ltx_font_bold">%</span></td>
<td id="S3.T4.3.1.1.4" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;">Examples</td>
</tr>
<tr id="S3.T4.3.1.2" class="ltx_tr">
<td id="S3.T4.3.1.2.1" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">Reasoning Error</td>
<td id="S3.T4.3.1.2.2" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">Mistakes due to incorrect reasoning steps or missing conditions.</td>
<td id="S3.T4.3.1.2.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">38%</td>
<td id="S3.T4.3.1.2.4" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">Ex. <a href="#LST5" title="Listing 5 ‣ F.2 Failure Cases ‣ Appendix F Examples ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">5</span></a>
</td>
</tr>
<tr id="S3.T4.3.1.3" class="ltx_tr">
<td id="S3.T4.3.1.3.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Hallucination</td>
<td id="S3.T4.3.1.3.2" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Fabrication of numbers or answers.</td>
<td id="S3.T4.3.1.3.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">5%</td>
<td id="S3.T4.3.1.3.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">Ex. <a href="#LST6" title="Listing 6 ‣ F.2 Failure Cases ‣ Appendix F Examples ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">6</span></a>
</td>
</tr>
<tr id="S3.T4.3.1.4" class="ltx_tr">
<td id="S3.T4.3.1.4.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Diagram Understanding</td>
<td id="S3.T4.3.1.4.2" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Misinterpretation of the input diagram.</td>
<td id="S3.T4.3.1.4.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">21%</td>
<td id="S3.T4.3.1.4.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">Ex. <a href="#LST7" title="Listing 7 ‣ F.2 Failure Cases ‣ Appendix F Examples ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">7</span></a>
</td>
</tr>
<tr id="S3.T4.3.1.5" class="ltx_tr">
<td id="S3.T4.3.1.5.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Inappropriate Tool Use</td>
<td id="S3.T4.3.1.5.2" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">
<table id="S3.T4.3.1.5.2.1" class="ltx_tabular ltx_align_middle">
<tr id="S3.T4.3.1.5.2.1.1" class="ltx_tr">
<td id="S3.T4.3.1.5.2.1.1.1" class="ltx_td ltx_nopad_r ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Incorrect use of external tools, especially when</td>
</tr>
<tr id="S3.T4.3.1.5.2.1.2" class="ltx_tr">
<td id="S3.T4.3.1.5.2.1.2.1" class="ltx_td ltx_nopad_r ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">the problem can’t be solved directly with libraries.</td>
</tr>
</table>
</td>
<td id="S3.T4.3.1.5.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">10%</td>
<td id="S3.T4.3.1.5.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">Ex. <a href="#LST8" title="Listing 8 ‣ F.2 Failure Cases ‣ Appendix F Examples ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">8</span></a>
</td>
</tr>
<tr id="S3.T4.3.1.6" class="ltx_tr">
<td id="S3.T4.3.1.6.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Syntax Error</td>
<td id="S3.T4.3.1.6.2" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Persistent syntax errors despite multiple correction attempts.</td>
<td id="S3.T4.3.1.6.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">9%</td>
<td id="S3.T4.3.1.6.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">Ex. <a href="#LST9" title="Listing 9 ‣ F.2 Failure Cases ‣ Appendix F Examples ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">9</span></a>
</td>
</tr>
<tr id="S3.T4.3.1.7" class="ltx_tr">
<td id="S3.T4.3.1.7.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Runtime Error</td>
<td id="S3.T4.3.1.7.2" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Errors during program execution, unresolved by retrying.</td>
<td id="S3.T4.3.1.7.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">9%</td>
<td id="S3.T4.3.1.7.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">Ex. <a href="#LST10" title="Listing 10 ‣ F.2 Failure Cases ‣ Appendix F Examples ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">10</span></a>
</td>
</tr>
<tr id="S3.T4.3.1.8" class="ltx_tr">
<td id="S3.T4.3.1.8.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Rationale-only Error</td>
<td id="S3.T4.3.1.8.2" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">
<table id="S3.T4.3.1.8.2.1" class="ltx_tabular ltx_align_middle">
<tr id="S3.T4.3.1.8.2.1.1" class="ltx_tr">
<td id="S3.T4.3.1.8.2.1.1.1" class="ltx_td ltx_nopad_r ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">Cannot be formalized into a program and the rationale is incorrect.</td>
</tr>
</table>
</td>
<td id="S3.T4.3.1.8.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">3%</td>
<td id="S3.T4.3.1.8.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">Ex. <a href="#LST11" title="Listing 11 ‣ F.2 Failure Cases ‣ Appendix F Examples ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">11</span></a>
</td>
</tr>
<tr id="S3.T4.3.1.9" class="ltx_tr">
<td id="S3.T4.3.1.9.1" class="ltx_td ltx_align_left ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">False Negative</td>
<td id="S3.T4.3.1.9.2" class="ltx_td ltx_align_left ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">Correct answers that don’t fully match the ground truth.</td>
<td id="S3.T4.3.1.9.3" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">5%</td>
<td id="S3.T4.3.1.9.4" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">Ex. <a href="#LST12" title="Listing 12 ‣ F.2 Failure Cases ‣ Appendix F Examples ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">12</span></a>
</td>
</tr>
</table>
</span></div>
</figure>
<div id="S3.SS6.p5" class="ltx_para ltx_noindent">
<p id="S3.SS6.p5.1" class="ltx_p"><span id="S3.SS6.p5.1.1" class="ltx_text ltx_font_bold">Remaining Challenges in Mathematical Reasoning for <span id="S3.SS6.p5.1.1.1" class="ltx_text ltx_font_smallcaps">ToRA</span></span>
To better understand the failure modes and remaining challenges,
we manually annotated 100 randomly selected trajectories from the MATH test set, identifying and categorizing their failure modes.
The results are shown in Table <a href="#S3.T4" title="Table 4 ‣ 3.6 Analysis ‣ 3 Experiments ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4</span></a>:
Primarily, incorrect reasoning steps constitute the primary source of errors for ToRA on complex math reasoning tasks (38%), with some hallucination issues also evident during problem interpretation and answer finalization (5%).
Secondly, the misinterpretation of input diagrams contributes significantly to the error rate (21%). This is particularly noticeable in Geometry, Precalculus, and Intermediate Algebra. The diagrams in the MATH dataset are usually detailed in text using the Asymptote language <cite class="ltx_cite ltx_citemacro_citep">(Hendrycks et al., <a href="#bib.bib12" title="" class="ltx_ref">2021</a>)</cite>, thus making it challenging for <span id="S3.SS6.p5.1.2" class="ltx_text ltx_font_smallcaps">ToRA</span> to comprehend diagrams purely from textual descriptions.
Thirdly, issues with tool usage include Inappropriate Tool Usage (10%), Syntax Error (9%), and Runtime Error (9%).
These problems frequently arise when <span id="S3.SS6.p5.1.3" class="ltx_text ltx_font_smallcaps">ToRA</span> fails to use tools correctly after several corrections or attempts. There are certain inputs that fail to formalize well as programs (3%), which require abstract reasoning rather than computation.
Finally, we also found that there are false negatives when using automatic indicators, i.e., correct predictions that are misjudged as wrong, but the proportion is relatively small (5%).</p>
</div>
</section>
</section>
<section id="S4" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">4 </span>Conclusion</h2>

<div id="S4.p1" class="ltx_para ltx_noindent">
<p id="S4.p1.1" class="ltx_p">This paper presents <span id="S4.p1.1.1" class="ltx_text ltx_font_smallcaps">ToRA</span>, a series of novel Tool-integrated Reasoning Agents that synergistically combines natural language rationale with program-based tool-use for mathematical problem solving. Our approach demonstrates the potential of integrating external tools in the reasoning process, enabling language models to effectively tackle complex quantitative tasks. <span id="S4.p1.1.2" class="ltx_text ltx_font_smallcaps">ToRA</span> achieves state-of-the-art performance on 10 diverse mathematical reasoning tasks, substantially outperforming existing rationale-based and program-based approaches. Furthermore, our systematic analysis of the benefits and remaining challenges of tool interaction provides valuable insights for future research, contributing to the development of more advanced and versatile reasoning agents.</p>
</div>
<section id="S4.SS0.SSSx1" class="ltx_subsubsection">
<h4 class="ltx_title ltx_title_subsubsection">Author Contributions</h4>

<div id="S4.SS0.SSSx1.p1" class="ltx_para ltx_noindent">
<p id="S4.SS0.SSSx1.p1.1" class="ltx_p">Zhibin Gou proposed the interleaved tool-use format of <span id="S4.SS0.SSSx1.p1.1.1" class="ltx_text ltx_font_smallcaps">ToRA</span> and curated <span id="S4.SS0.SSSx1.p1.1.2" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span> dataset, implemented the training and evaluation pipeline, conducted experiments and analysis on all datasets, implemented baselines, and was a main contributor to the paper writing.
Zhihong Shao proposed the project, conducted preliminary experiments, proposed and implemented the training and evaluation pipelines, proposed and trained all <span id="S4.SS0.SSSx1.p1.1.3" class="ltx_text ltx_font_smallcaps">ToRA</span> models with output space shaping as well as <span id="S4.SS0.SSSx1.p1.1.4" class="ltx_text ltx_font_smallcaps">ToRA</span> variants in the ablation study, designed and oversaw experimental analysis, and contributed to many parts of the paper writing.
Yeyun Gong, Yelong Shen, Yujiu Yang, Minlie Huang, Nan Duan, and Weizhu Chen provided research mentorship, oversaw project coordination, and advised and contributed to many parts of the writing.</p>
</div>
</section>
<section id="S4.SS0.SSSx2" class="ltx_subsubsection">
<h4 class="ltx_title ltx_title_subsubsection">Acknowledgments</h4>

<div id="S4.SS0.SSSx2.p1" class="ltx_para ltx_noindent">
<p id="S4.SS0.SSSx2.p1.1" class="ltx_p">Zhibin Gou and Yujiu Yang were supported by the National Natural Science Foundation of China (Grant No. U1903213) and the Shenzhen Science and Technology Program (JSGG20220831110203007).
Zhihong Shao and Minlie Huang were supported by the NSFC projects (Key project with No. 61936010 ), and were also supported by the National Science Foundation for Distinguished Young Scholars (with No. 62125604).</p>
</div>
<div class="ltx_pagination ltx_role_newpage"></div>
</section>
</section>
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<div class="ltx_pagination ltx_role_newpage"></div>
<nav class="ltx_TOC ltx_list_toc ltx_toc_toc"><h6 class="ltx_title ltx_title_contents">Contents</h6>
<ol class="ltx_toclist">
<li class="ltx_tocentry ltx_tocentry_section"><a href="#S1" title="In \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li>
<li class="ltx_tocentry ltx_tocentry_section">
<a href="#S2" title="In \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span><span class="ltx_text ltx_font_smallcaps">ToRA</span>: Tool-Integrated Agents for Mathematical Reasoning</span></a>
<ol class="ltx_toclist ltx_toclist_section">
<li class="ltx_tocentry ltx_tocentry_subsection"><a href="#S2.SS1" title="In 2 ToRA: Tool-Integrated Agents for Mathematical Reasoning ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Overview</span></a></li>
<li class="ltx_tocentry ltx_tocentry_subsection"><a href="#S2.SS2" title="In 2 ToRA: Tool-Integrated Agents for Mathematical Reasoning ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Collecting Interactive Tool-Use Trajectories</span></a></li>
<li class="ltx_tocentry ltx_tocentry_subsection"><a href="#S2.SS3" title="In 2 ToRA: Tool-Integrated Agents for Mathematical Reasoning ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Training</span></a></li>
</ol>
</li>
<li class="ltx_tocentry ltx_tocentry_section">
<a href="#S3" title="In \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Experiments</span></a>
<ol class="ltx_toclist ltx_toclist_section">
<li class="ltx_tocentry ltx_tocentry_subsection"><a href="#S3.SS1" title="In 3 Experiments ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Implementation Details</span></a></li>
<li class="ltx_tocentry ltx_tocentry_subsection"><a href="#S3.SS2" title="In 3 Experiments ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Evaluation Setup</span></a></li>
<li class="ltx_tocentry ltx_tocentry_subsection"><a href="#S3.SS3" title="In 3 Experiments ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>Baselines</span></a></li>
<li class="ltx_tocentry ltx_tocentry_subsection"><a href="#S3.SS4" title="In 3 Experiments ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.4 </span>Main Results</span></a></li>
<li class="ltx_tocentry ltx_tocentry_subsection">
<a href="#S3.SS5" title="In 3 Experiments ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.5 </span>Ablation Study</span></a>
<ol class="ltx_toclist ltx_toclist_subsection">
<li class="ltx_tocentry ltx_tocentry_subsubsection"><a href="#S3.SS5.SSS1" title="In 3.5 Ablation Study ‣ 3 Experiments ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.5.1 </span>Comparisons of Formatting</span></a></li>
<li class="ltx_tocentry ltx_tocentry_subsubsection"><a href="#S3.SS5.SSS2" title="In 3.5 Ablation Study ‣ 3 Experiments ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.5.2 </span>Effects of Output Space Shaping</span></a></li>
</ol>
</li>
<li class="ltx_tocentry ltx_tocentry_subsection"><a href="#S3.SS6" title="In 3 Experiments ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.6 </span>Analysis</span></a></li>
</ol>
</li>
<li class="ltx_tocentry ltx_tocentry_section"><a href="#S4" title="In \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Conclusion</span></a></li>
<li class="ltx_tocentry ltx_tocentry_appendix"><a href="#A1" title="In \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>Related Works</span></a></li>
<li class="ltx_tocentry ltx_tocentry_appendix"><a href="#A2" title="In \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B </span>Evaluation Datasets</span></a></li>
<li class="ltx_tocentry ltx_tocentry_appendix">
<a href="#A3" title="In \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C </span>Additional Experiments and Analysis</span></a>
<ol class="ltx_toclist ltx_toclist_appendix">
<li class="ltx_tocentry ltx_tocentry_subsection"><a href="#A3.SS1" title="In Appendix C Additional Experiments and Analysis ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C.1 </span>Accuracies of Closed-Source Models on MATH</span></a></li>
<li class="ltx_tocentry ltx_tocentry_subsection"><a href="#A3.SS2" title="In Appendix C Additional Experiments and Analysis ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C.2 </span>Effects of # Valid Trajectories for Output Space Shaping</span></a></li>
<li class="ltx_tocentry ltx_tocentry_subsection"><a href="#A3.SS3" title="In Appendix C Additional Experiments and Analysis ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C.3 </span>Impact of Output Space Shaping in Relation to Question Difficulty</span></a></li>
</ol>
</li>
<li class="ltx_tocentry ltx_tocentry_appendix"><a href="#A4" title="In \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D </span>Detailed Information of <span class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span></span></a></li>
<li class="ltx_tocentry ltx_tocentry_appendix"><a href="#A5" title="In \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">E </span>Prompts</span></a></li>
<li class="ltx_tocentry ltx_tocentry_appendix">
<a href="#A6" title="In \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">F </span>Examples</span></a>
<ol class="ltx_toclist ltx_toclist_appendix">
<li class="ltx_tocentry ltx_tocentry_subsection"><a href="#A6.SS1" title="In Appendix F Examples ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">F.1 </span>Success Cases</span></a></li>
<li class="ltx_tocentry ltx_tocentry_subsection"><a href="#A6.SS2" title="In Appendix F Examples ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">F.2 </span>Failure Cases</span></a></li>
</ol>
</li>
</ol></nav>
<section id="A1" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix A </span>Related Works</h2>

<div id="A1.p1" class="ltx_para ltx_noindent">
<p id="A1.p1.1" class="ltx_p"><span id="A1.p1.1.1" class="ltx_text ltx_font_bold">Mathematical Reasoning</span> Recent research has greatly improved reasoning in LLMs with step-by-step natural language reasoning <cite class="ltx_cite ltx_citemacro_citep">(Polu &amp; Sutskever, <a href="#bib.bib35" title="" class="ltx_ref">2020</a>; Wei et al., <a href="#bib.bib45" title="" class="ltx_ref">2022</a>; Zhou et al., <a href="#bib.bib52" title="" class="ltx_ref">2023b</a>; Zhu et al., <a href="#bib.bib53" title="" class="ltx_ref">2023</a>; Huang et al., <a href="#bib.bib16" title="" class="ltx_ref">2022</a>; Liang et al., <a href="#bib.bib21" title="" class="ltx_ref">2023</a>)</cite>.
However, natural language reasoning struggles with complex computations and symbolic manipulations.
To overcome the limitations, recent research has exploited tools like calculators <cite class="ltx_cite ltx_citemacro_citep">(Cobbe et al., <a href="#bib.bib6" title="" class="ltx_ref">2021</a>; Shao et al., <a href="#bib.bib39" title="" class="ltx_ref">2022</a>)</cite>, code interpreters <cite class="ltx_cite ltx_citemacro_citep">(Mishra et al., <a href="#bib.bib27" title="" class="ltx_ref">2022</a>)</cite>, and symbolic solvers <cite class="ltx_cite ltx_citemacro_citep">(Zhang et al., <a href="#bib.bib50" title="" class="ltx_ref">2023</a>)</cite>.
Program-based methods <cite class="ltx_cite ltx_citemacro_citep">(Gao et al., <a href="#bib.bib10" title="" class="ltx_ref">2022</a>; Chen et al., <a href="#bib.bib5" title="" class="ltx_ref">2022</a>; Shao et al., <a href="#bib.bib40" title="" class="ltx_ref">2023a</a>)</cite> transform reasoning tasks into program synthesis tasks, thus offering complementary advantages over natural language reasoning, but they face challenges in nuanced reasoning, planning, and error handling <cite class="ltx_cite ltx_citemacro_citep">(Gou et al., <a href="#bib.bib11" title="" class="ltx_ref">2023</a>)</cite>, where natural language reasoning should be more suitable.</p>
</div>
<div id="A1.p2" class="ltx_para ltx_noindent">
<p id="A1.p2.1" class="ltx_p"><span id="A1.p2.1.1" class="ltx_text ltx_font_bold">Tool-Augmented Language Models</span>
Augmenting LLMs with tools can largely alleviate LLMs’ limitations and improve reasoning and generation performance <cite class="ltx_cite ltx_citemacro_citep">(Parisi et al., <a href="#bib.bib31" title="" class="ltx_ref">2022</a>; Mialon et al., <a href="#bib.bib25" title="" class="ltx_ref">2023</a>; Yao et al., <a href="#bib.bib46" title="" class="ltx_ref">2023</a>)</cite>.
Recent work demonstrates the benefits of integrating retrievers <cite class="ltx_cite ltx_citemacro_citep">(Borgeaud et al., <a href="#bib.bib2" title="" class="ltx_ref">2022</a>; Shao et al., <a href="#bib.bib41" title="" class="ltx_ref">2023b</a>)</cite>, search engines <cite class="ltx_cite ltx_citemacro_citep">(Nakano et al., <a href="#bib.bib28" title="" class="ltx_ref">2021</a>)</cite>, and multi-tool approaches <cite class="ltx_cite ltx_citemacro_citep">(Schick et al., <a href="#bib.bib38" title="" class="ltx_ref">2023</a>; Paranjape et al., <a href="#bib.bib30" title="" class="ltx_ref">2023</a>; Gou et al., <a href="#bib.bib11" title="" class="ltx_ref">2023</a>)</cite> to improve generation.</p>
</div>
<div id="A1.p3" class="ltx_para ltx_noindent">
<p id="A1.p3.1" class="ltx_p"><span id="A1.p3.1.1" class="ltx_text ltx_font_bold">Knowledge Distillation</span>
Knowledge distillation (KD) transfers knowledge from teacher models to student models <cite class="ltx_cite ltx_citemacro_citep">(Buciluǎ et al., <a href="#bib.bib4" title="" class="ltx_ref">2006</a>; Hinton et al., <a href="#bib.bib13" title="" class="ltx_ref">2015</a>)</cite>. Using LLM-generated trajectories for fine-tuning is a form of KD <cite class="ltx_cite ltx_citemacro_citep">(Fu et al., <a href="#bib.bib9" title="" class="ltx_ref">2023</a>; Taori et al., <a href="#bib.bib42" title="" class="ltx_ref">2023</a>; Peng et al., <a href="#bib.bib34" title="" class="ltx_ref">2023</a>; Ho et al., <a href="#bib.bib14" title="" class="ltx_ref">2023</a>)</cite>.
Our proposed <span id="A1.p3.1.2" class="ltx_text ltx_font_smallcaps">ToRA</span> shows that learning interactive tool-use trajectories is a promising direction to adapt language models to reasoning tasks.</p>
</div>
</section>
<section id="A2" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix B </span>Evaluation Datasets</h2>

<figure id="A2.T5" class="ltx_table">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 5: </span>Statistics and examples of the 10 evaluation datasets. In the main result table, we present the average accuracy of SingleEq, SingleOp, AddSub, and MultiArith under the collective name MAWPS.</figcaption>
<div id="A2.T5.2" class="ltx_inline-block ltx_transformed_outer" style="width:433.6pt;height:718.9pt;vertical-align:-1.1pt;"><span class="ltx_transformed_inner" style="transform:translate(12.7pt,-21.0pt) scale(1.06199848124275,1.06199848124275) ;">
<table id="A2.T5.2.2" class="ltx_tabular ltx_align_middle">
<tr id="A2.T5.2.2.3" class="ltx_tr">
<td id="A2.T5.2.2.3.1" class="ltx_td ltx_align_justify ltx_align_middle ltx_border_tt" style="width:76.8pt;padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.3.1.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.3.1.1.1" class="ltx_p">Dataset</span>
</span>
</td>
<td id="A2.T5.2.2.3.2" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;">OOD?</td>
<td id="A2.T5.2.2.3.3" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;">#Samples</td>
<td id="A2.T5.2.2.3.4" class="ltx_td ltx_align_justify ltx_align_top ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.3.4.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.3.4.1.1" class="ltx_p" style="width:227.6pt;">Example Problem</span>
</span>
</td>
</tr>
<tr id="A2.T5.2.2.4" class="ltx_tr">
<td id="A2.T5.2.2.4.1" class="ltx_td ltx_align_justify ltx_align_middle ltx_border_t" style="width:76.8pt;padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.4.1.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.4.1.1.1" class="ltx_p">
<span id="A2.T5.2.2.4.1.1.1.1" class="ltx_inline-block ltx_parbox ltx_align_top" style="width:76.8pt;">
<span id="A2.T5.2.2.4.1.1.1.1.1" class="ltx_p">GSM8k <cite class="ltx_cite ltx_citemacro_citep">(Cobbe et al., <a href="#bib.bib6" title="" class="ltx_ref">2021</a>)</cite></span>
</span></span>
</span>
</td>
<td id="A2.T5.2.2.4.2" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.4.2.1" class="ltx_text">IND</span></td>
<td id="A2.T5.2.2.4.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.4.3.1" class="ltx_text">1319</span></td>
<td id="A2.T5.2.2.4.4" class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.4.4.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.4.4.1.1" class="ltx_p" style="width:227.6pt;">The ice cream parlor was offering a deal, buy 2 scoops of ice cream, get 1 scoop free. Each scoop cost $1.50. If Erin had $6.00, how many scoops of ice cream should she buy?</span>
</span>
</td>
</tr>
<tr id="A2.T5.2.2.2" class="ltx_tr">
<td id="A2.T5.2.2.2.3" class="ltx_td ltx_align_justify ltx_align_middle ltx_border_t" style="width:76.8pt;padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.2.3.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.2.3.1.1" class="ltx_p">
<span id="A2.T5.2.2.2.3.1.1.1" class="ltx_inline-block ltx_parbox ltx_align_top" style="width:76.8pt;">
<span id="A2.T5.2.2.2.3.1.1.1.1" class="ltx_p">MATH <cite class="ltx_cite ltx_citemacro_citep">(Hendrycks et al., <a href="#bib.bib12" title="" class="ltx_ref">2021</a>)</cite></span>
</span></span>
</span>
</td>
<td id="A2.T5.2.2.2.4" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.2.4.1" class="ltx_text">IND</span></td>
<td id="A2.T5.2.2.2.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.2.5.1" class="ltx_text">5000</span></td>
<td id="A2.T5.2.2.2.2" class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.2.2.2" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.2.2.2.2" class="ltx_p" style="width:227.6pt;">For a constant <math id="A2.T5.1.1.1.1.1.1.m1.1" class="ltx_Math" alttext="c," display="inline"><semantics id="A2.T5.1.1.1.1.1.1.m1.1a"><mrow id="A2.T5.1.1.1.1.1.1.m1.1.2.2"><mi id="A2.T5.1.1.1.1.1.1.m1.1.1" xref="A2.T5.1.1.1.1.1.1.m1.1.1.cmml">c</mi><mo id="A2.T5.1.1.1.1.1.1.m1.1.2.2.1">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.T5.1.1.1.1.1.1.m1.1b"><ci id="A2.T5.1.1.1.1.1.1.m1.1.1.cmml" xref="A2.T5.1.1.1.1.1.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.T5.1.1.1.1.1.1.m1.1c">c,</annotation></semantics></math> in cylindrical coordinates <math id="A2.T5.2.2.2.2.2.2.m2.4" class="ltx_Math" alttext="(r,\theta,z)," display="inline"><semantics id="A2.T5.2.2.2.2.2.2.m2.4a"><mrow id="A2.T5.2.2.2.2.2.2.m2.4.4.1"><mrow id="A2.T5.2.2.2.2.2.2.m2.4.4.1.1.2" xref="A2.T5.2.2.2.2.2.2.m2.4.4.1.1.1.cmml"><mo stretchy="false" id="A2.T5.2.2.2.2.2.2.m2.4.4.1.1.2.1" xref="A2.T5.2.2.2.2.2.2.m2.4.4.1.1.1.cmml">(</mo><mi id="A2.T5.2.2.2.2.2.2.m2.1.1" xref="A2.T5.2.2.2.2.2.2.m2.1.1.cmml">r</mi><mo id="A2.T5.2.2.2.2.2.2.m2.4.4.1.1.2.2" xref="A2.T5.2.2.2.2.2.2.m2.4.4.1.1.1.cmml">,</mo><mi id="A2.T5.2.2.2.2.2.2.m2.2.2" xref="A2.T5.2.2.2.2.2.2.m2.2.2.cmml">θ</mi><mo id="A2.T5.2.2.2.2.2.2.m2.4.4.1.1.2.3" xref="A2.T5.2.2.2.2.2.2.m2.4.4.1.1.1.cmml">,</mo><mi id="A2.T5.2.2.2.2.2.2.m2.3.3" xref="A2.T5.2.2.2.2.2.2.m2.3.3.cmml">z</mi><mo stretchy="false" id="A2.T5.2.2.2.2.2.2.m2.4.4.1.1.2.4" xref="A2.T5.2.2.2.2.2.2.m2.4.4.1.1.1.cmml">)</mo></mrow><mo id="A2.T5.2.2.2.2.2.2.m2.4.4.1.2">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.T5.2.2.2.2.2.2.m2.4b"><vector id="A2.T5.2.2.2.2.2.2.m2.4.4.1.1.1.cmml" xref="A2.T5.2.2.2.2.2.2.m2.4.4.1.1.2"><ci id="A2.T5.2.2.2.2.2.2.m2.1.1.cmml" xref="A2.T5.2.2.2.2.2.2.m2.1.1">𝑟</ci><ci id="A2.T5.2.2.2.2.2.2.m2.2.2.cmml" xref="A2.T5.2.2.2.2.2.2.m2.2.2">𝜃</ci><ci id="A2.T5.2.2.2.2.2.2.m2.3.3.cmml" xref="A2.T5.2.2.2.2.2.2.m2.3.3">𝑧</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="A2.T5.2.2.2.2.2.2.m2.4c">(r,\theta,z),</annotation></semantics></math> find the shape described by the equation 
<span id="A2.Ex1" class="ltx_equation ltx_eqn_table">

<span><span class="ltx_equation ltx_eqn_row ltx_align_baseline">
<span class="ltx_eqn_cell ltx_eqn_center_padleft"></span>
<span class="ltx_eqn_cell ltx_align_center"><math id="A2.Ex1.m1.1" class="ltx_Math" alttext="z=c." display="block"><semantics id="A2.Ex1.m1.1a"><mrow id="A2.Ex1.m1.1.1.1" xref="A2.Ex1.m1.1.1.1.1.cmml"><mrow id="A2.Ex1.m1.1.1.1.1" xref="A2.Ex1.m1.1.1.1.1.cmml"><mi id="A2.Ex1.m1.1.1.1.1.2" xref="A2.Ex1.m1.1.1.1.1.2.cmml">z</mi><mo id="A2.Ex1.m1.1.1.1.1.1" xref="A2.Ex1.m1.1.1.1.1.1.cmml">=</mo><mi id="A2.Ex1.m1.1.1.1.1.3" xref="A2.Ex1.m1.1.1.1.1.3.cmml">c</mi></mrow><mo lspace="0em" id="A2.Ex1.m1.1.1.1.2" xref="A2.Ex1.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex1.m1.1b"><apply id="A2.Ex1.m1.1.1.1.1.cmml" xref="A2.Ex1.m1.1.1.1"><eq id="A2.Ex1.m1.1.1.1.1.1.cmml" xref="A2.Ex1.m1.1.1.1.1.1"></eq><ci id="A2.Ex1.m1.1.1.1.1.2.cmml" xref="A2.Ex1.m1.1.1.1.1.2">𝑧</ci><ci id="A2.Ex1.m1.1.1.1.1.3.cmml" xref="A2.Ex1.m1.1.1.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex1.m1.1c">z=c.</annotation></semantics></math></span>
<span class="ltx_eqn_cell ltx_eqn_center_padright"></span></span></span>
</span>(A) Line (B) Circle (C) Plane (D) Sphere (E) Cylinder (F) Cone. Enter the letter of the correct option.</span>
</span>
</td>
</tr>
<tr id="A2.T5.2.2.5" class="ltx_tr">
<td id="A2.T5.2.2.5.1" class="ltx_td ltx_align_justify ltx_align_middle ltx_border_t" style="width:76.8pt;padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.5.1.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.5.1.1.1" class="ltx_p">
<span id="A2.T5.2.2.5.1.1.1.1" class="ltx_inline-block ltx_parbox ltx_align_top" style="width:76.8pt;">
<span id="A2.T5.2.2.5.1.1.1.1.1" class="ltx_p">GSM-Hard <cite class="ltx_cite ltx_citemacro_citep">(Gao et al., <a href="#bib.bib10" title="" class="ltx_ref">2022</a>)</cite></span>
</span></span>
</span>
</td>
<td id="A2.T5.2.2.5.2" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.5.2.1" class="ltx_text">OOD</span></td>
<td id="A2.T5.2.2.5.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.5.3.1" class="ltx_text">1319</span></td>
<td id="A2.T5.2.2.5.4" class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.5.4.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.5.4.1.1" class="ltx_p" style="width:227.6pt;">Jean has 30 lollipops. Jean eats 8714250 of the lollipops. With the remaining lollipops, Jean wants to package 8714250 lollipops in one bag. How many bags can Jean fill?</span>
</span>
</td>
</tr>
<tr id="A2.T5.2.2.6" class="ltx_tr">
<td id="A2.T5.2.2.6.1" class="ltx_td ltx_align_justify ltx_align_middle ltx_border_t" style="width:76.8pt;padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.6.1.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.6.1.1.1" class="ltx_p">
<span id="A2.T5.2.2.6.1.1.1.1" class="ltx_inline-block ltx_parbox ltx_align_top" style="width:76.8pt;">
<span id="A2.T5.2.2.6.1.1.1.1.1" class="ltx_p">SVAMP <cite class="ltx_cite ltx_citemacro_citep">(Patel et al., <a href="#bib.bib32" title="" class="ltx_ref">2021</a>)</cite></span>
</span></span>
</span>
</td>
<td id="A2.T5.2.2.6.2" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.6.2.1" class="ltx_text">OOD</span></td>
<td id="A2.T5.2.2.6.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.6.3.1" class="ltx_text">1000</span></td>
<td id="A2.T5.2.2.6.4" class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.6.4.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.6.4.1.1" class="ltx_p" style="width:227.6pt;">During summer break 819058 kids from Lawrence county go to camp and the other 668278 kids stay home. How many more kids spent their summer break at the camp compared to those who stayed home?</span>
</span>
</td>
</tr>
<tr id="A2.T5.2.2.7" class="ltx_tr">
<td id="A2.T5.2.2.7.1" class="ltx_td ltx_align_justify ltx_align_middle ltx_border_t" style="width:76.8pt;padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.7.1.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.7.1.1.1" class="ltx_p">
<span id="A2.T5.2.2.7.1.1.1.1" class="ltx_inline-block ltx_parbox ltx_align_top" style="width:76.8pt;">
<span id="A2.T5.2.2.7.1.1.1.1.1" class="ltx_p">ASDiv <cite class="ltx_cite ltx_citemacro_citep">(Miao et al., <a href="#bib.bib26" title="" class="ltx_ref">2020</a>)</cite></span>
</span></span>
</span>
</td>
<td id="A2.T5.2.2.7.2" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.7.2.1" class="ltx_text">OOD</span></td>
<td id="A2.T5.2.2.7.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.7.3.1" class="ltx_text">2215</span></td>
<td id="A2.T5.2.2.7.4" class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.7.4.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.7.4.1.1" class="ltx_p" style="width:227.6pt;">Mrs. Hilt saw an iPod for sale. The price tag said the iPod cost $128, but a sign announced that it was on sale for "35% off." How much would the iPod cost after the discount?</span>
</span>
</td>
</tr>
<tr id="A2.T5.2.2.8" class="ltx_tr">
<td id="A2.T5.2.2.8.1" class="ltx_td ltx_align_justify ltx_align_middle ltx_border_t" style="width:76.8pt;padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.8.1.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.8.1.1.1" class="ltx_p">
<span id="A2.T5.2.2.8.1.1.1.1" class="ltx_inline-block ltx_parbox ltx_align_top" style="width:76.8pt;">
<span id="A2.T5.2.2.8.1.1.1.1.1" class="ltx_p">TabMWP <cite class="ltx_cite ltx_citemacro_citep">(Lu et al., <a href="#bib.bib23" title="" class="ltx_ref">2023</a>)</cite></span>
</span></span>
</span>
</td>
<td id="A2.T5.2.2.8.2" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.8.2.1" class="ltx_text">OOD</span></td>
<td id="A2.T5.2.2.8.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.8.3.1" class="ltx_text">1000</span></td>
<td id="A2.T5.2.2.8.4" class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.8.4.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.8.4.1.1" class="ltx_p" style="width:227.6pt;">
<span id="A2.T5.2.2.8.4.1.1.1" class="ltx_tabular ltx_minipage ltx_align_middle" style="width:79.5pt;">
<span id="A2.T5.2.2.8.4.1.1.1.1" class="ltx_tr">
<span id="A2.T5.2.2.8.4.1.1.1.1.1" class="ltx_td ltx_align_left ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;">Stem</span>
<span id="A2.T5.2.2.8.4.1.1.1.1.2" class="ltx_td ltx_align_left ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;">Leaf</span></span>
<span id="A2.T5.2.2.8.4.1.1.1.2" class="ltx_tr">
<span id="A2.T5.2.2.8.4.1.1.1.2.1" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">2</span>
<span id="A2.T5.2.2.8.4.1.1.1.2.2" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">3, 6, 7, 8, 8</span></span>
<span id="A2.T5.2.2.8.4.1.1.1.3" class="ltx_tr">
<span id="A2.T5.2.2.8.4.1.1.1.3.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">3</span>
<span id="A2.T5.2.2.8.4.1.1.1.3.2" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">0, 7, 9</span></span>
<span id="A2.T5.2.2.8.4.1.1.1.4" class="ltx_tr">
<span id="A2.T5.2.2.8.4.1.1.1.4.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">4</span>
<span id="A2.T5.2.2.8.4.1.1.1.4.2" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">1, 5</span></span>
<span id="A2.T5.2.2.8.4.1.1.1.5" class="ltx_tr">
<span id="A2.T5.2.2.8.4.1.1.1.5.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">5</span>
<span id="A2.T5.2.2.8.4.1.1.1.5.2" class="ltx_td" style="padding-left:4.0pt;padding-right:4.0pt;"></span></span>
<span id="A2.T5.2.2.8.4.1.1.1.6" class="ltx_tr">
<span id="A2.T5.2.2.8.4.1.1.1.6.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">6</span>
<span id="A2.T5.2.2.8.4.1.1.1.6.2" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">2, 3, 3, 4, 8, 8</span></span>
<span id="A2.T5.2.2.8.4.1.1.1.7" class="ltx_tr">
<span id="A2.T5.2.2.8.4.1.1.1.7.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">7</span>
<span id="A2.T5.2.2.8.4.1.1.1.7.2" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">3, 4, 4, 7, 9</span></span>
<span id="A2.T5.2.2.8.4.1.1.1.8" class="ltx_tr">
<span id="A2.T5.2.2.8.4.1.1.1.8.1" class="ltx_td ltx_align_left ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">8</span>
<span id="A2.T5.2.2.8.4.1.1.1.8.2" class="ltx_td ltx_align_left ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">5, 5</span></span>
</span>
         

<span id="A2.T5.2.2.8.4.1.1.2" class="ltx_inline-block ltx_minipage ltx_align_middle" style="width:99.4pt;">
<span id="A2.T5.2.2.8.4.1.1.2.1" class="ltx_p">Read the table regarding “eight lifting results (lbs)”.
Mr. Morrison, a P.E. teacher, wrote down how much weight each of his students could lift. How many people lifted at least 28 pounds?</span>
</span></span>
</span>
</td>
</tr>
<tr id="A2.T5.2.2.9" class="ltx_tr">
<td id="A2.T5.2.2.9.1" class="ltx_td ltx_align_justify ltx_align_middle ltx_border_t" style="width:76.8pt;padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.9.1.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.9.1.1.1" class="ltx_p">
<span id="A2.T5.2.2.9.1.1.1.1" class="ltx_inline-block ltx_parbox ltx_align_top" style="width:76.8pt;">
<span id="A2.T5.2.2.9.1.1.1.1.1" class="ltx_p">SingleEq <cite class="ltx_cite ltx_citemacro_citep">(Koncel-Kedziorski et al., <a href="#bib.bib18" title="" class="ltx_ref">2016</a>)</cite></span>
</span></span>
</span>
</td>
<td id="A2.T5.2.2.9.2" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.9.2.1" class="ltx_text">OOD</span></td>
<td id="A2.T5.2.2.9.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.9.3.1" class="ltx_text">508</span></td>
<td id="A2.T5.2.2.9.4" class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.9.4.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.9.4.1.1" class="ltx_p" style="width:227.6pt;">Alyssa’s dog had puppies. She gave 7 to her friends.She now has 5 puppies left. How many puppies did she have to start with?</span>
</span>
</td>
</tr>
<tr id="A2.T5.2.2.10" class="ltx_tr">
<td id="A2.T5.2.2.10.1" class="ltx_td ltx_align_justify ltx_align_middle ltx_border_t" style="width:76.8pt;padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.10.1.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.10.1.1.1" class="ltx_p">
<span id="A2.T5.2.2.10.1.1.1.1" class="ltx_inline-block ltx_parbox ltx_align_top" style="width:76.8pt;">
<span id="A2.T5.2.2.10.1.1.1.1.1" class="ltx_p">SingleOp <cite class="ltx_cite ltx_citemacro_citep">(Koncel-Kedziorski et al., <a href="#bib.bib18" title="" class="ltx_ref">2016</a>)</cite></span>
</span></span>
</span>
</td>
<td id="A2.T5.2.2.10.2" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.10.2.1" class="ltx_text">OOD</span></td>
<td id="A2.T5.2.2.10.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.10.3.1" class="ltx_text">562</span></td>
<td id="A2.T5.2.2.10.4" class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.10.4.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.10.4.1.1" class="ltx_p" style="width:227.6pt;">Rachel removes 47 bottle caps from a jar. There were originally 87 bottle caps in the jar. How many bottle caps are left in the jar?</span>
</span>
</td>
</tr>
<tr id="A2.T5.2.2.11" class="ltx_tr">
<td id="A2.T5.2.2.11.1" class="ltx_td ltx_align_justify ltx_align_middle ltx_border_t" style="width:76.8pt;padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.11.1.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.11.1.1.1" class="ltx_p">
<span id="A2.T5.2.2.11.1.1.1.1" class="ltx_inline-block ltx_parbox ltx_align_top" style="width:76.8pt;">
<span id="A2.T5.2.2.11.1.1.1.1.1" class="ltx_p">AddSub <cite class="ltx_cite ltx_citemacro_citep">(Koncel-Kedziorski et al., <a href="#bib.bib18" title="" class="ltx_ref">2016</a>)</cite></span>
</span></span>
</span>
</td>
<td id="A2.T5.2.2.11.2" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.11.2.1" class="ltx_text">OOD</span></td>
<td id="A2.T5.2.2.11.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A2.T5.2.2.11.3.1" class="ltx_text">395</span></td>
<td id="A2.T5.2.2.11.4" class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.11.4.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.11.4.1.1" class="ltx_p" style="width:227.6pt;">Sam went to 14 football games this year. He went to 29 games last year. How many football games did Sam go to in all?</span>
</span>
</td>
</tr>
<tr id="A2.T5.2.2.12" class="ltx_tr">
<td id="A2.T5.2.2.12.1" class="ltx_td ltx_align_justify ltx_align_middle ltx_border_bb ltx_border_t" style="width:76.8pt;padding-left:4.0pt;padding-right:4.0pt;" rowspan="3">
<span id="A2.T5.2.2.12.1.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.12.1.1.1" class="ltx_p">
<span id="A2.T5.2.2.12.1.1.1.1" class="ltx_inline-block ltx_parbox ltx_align_top" style="width:76.8pt;">
<span id="A2.T5.2.2.12.1.1.1.1.1" class="ltx_p">MultArith <cite class="ltx_cite ltx_citemacro_citep">(Koncel-Kedziorski et al., <a href="#bib.bib18" title="" class="ltx_ref">2016</a>)</cite></span>
</span></span>
</span>
</td>
<td id="A2.T5.2.2.12.2" class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;" rowspan="3"><span id="A2.T5.2.2.12.2.1" class="ltx_text">OOD</span></td>
<td id="A2.T5.2.2.12.3" class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;" rowspan="3"><span id="A2.T5.2.2.12.3.1" class="ltx_text">600</span></td>
<td id="A2.T5.2.2.12.4" class="ltx_td ltx_align_justify ltx_align_top ltx_border_bb ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="A2.T5.2.2.12.4.1" class="ltx_inline-block ltx_align_top">
<span id="A2.T5.2.2.12.4.1.1" class="ltx_p" style="width:227.6pt;">Paige had 43 math problems and 12 science problems for homework. If she finished 44 of the problems at school, how many problems did she have to do for homework?</span>
</span>
</td>
</tr>
</table>
</span></div>
</figure>
<div id="A2.p1" class="ltx_para ltx_noindent">
<p id="A2.p1.1" class="ltx_p">We present statistics and examples of the ten evaluation datasets in Table <a href="#A2.T5" title="Table 5 ‣ Appendix B Evaluation Datasets ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">5</span></a>.</p>
</div>
</section>
<section id="A3" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix C </span>Additional Experiments and Analysis</h2>

<section id="A3.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">C.1 </span>Accuracies of Closed-Source Models on MATH</h3>

<figure id="A3.T6" class="ltx_table">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 6: </span>
Accuracies of ChatGPT and GPT-4 on the MATH dataset, with breakdown w.r.t. different mathematical subjects.
We apply PAL prompting and the Tool-integrated Reasoning method used by <span id="A3.T6.4.1" class="ltx_text ltx_font_smallcaps">ToRA</span> to the two closed-source models.
</figcaption>
<div id="A3.T6.2" class="ltx_inline-block ltx_align_center ltx_transformed_outer" style="width:433.6pt;height:116.7pt;vertical-align:-0.7pt;"><span class="ltx_transformed_inner" style="transform:translate(-96.0pt,25.7pt) scale(0.693174684253967,0.693174684253967) ;">
<table id="A3.T6.2.2" class="ltx_tabular ltx_align_middle">
<tr id="A3.T6.2.2.3" class="ltx_tr">
<td id="A3.T6.2.2.3.1" class="ltx_td ltx_align_left ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A3.T6.2.2.3.1.1" class="ltx_text ltx_font_bold">Model</span></td>
<td id="A3.T6.2.2.3.2" class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A3.T6.2.2.3.2.1" class="ltx_text ltx_font_bold">Tool</span></td>
<td id="A3.T6.2.2.3.3" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;">
<table id="A3.T6.2.2.3.3.1" class="ltx_tabular ltx_align_middle">
<tr id="A3.T6.2.2.3.3.1.1" class="ltx_tr">
<td id="A3.T6.2.2.3.3.1.1.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A3.T6.2.2.3.3.1.1.1.1" class="ltx_text ltx_font_bold">Intermediate</span></td>
</tr>
<tr id="A3.T6.2.2.3.3.1.2" class="ltx_tr">
<td id="A3.T6.2.2.3.3.1.2.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A3.T6.2.2.3.3.1.2.1.1" class="ltx_text ltx_font_bold">Algebra</span></td>
</tr>
</table>
</td>
<td id="A3.T6.2.2.3.4" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A3.T6.2.2.3.4.1" class="ltx_text ltx_font_bold">Precalculus</span></td>
<td id="A3.T6.2.2.3.5" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A3.T6.2.2.3.5.1" class="ltx_text ltx_font_bold">Geometry</span></td>
<td id="A3.T6.2.2.3.6" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;">
<table id="A3.T6.2.2.3.6.1" class="ltx_tabular ltx_align_middle">
<tr id="A3.T6.2.2.3.6.1.1" class="ltx_tr">
<td id="A3.T6.2.2.3.6.1.1.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A3.T6.2.2.3.6.1.1.1.1" class="ltx_text ltx_font_bold">Number</span></td>
</tr>
<tr id="A3.T6.2.2.3.6.1.2" class="ltx_tr">
<td id="A3.T6.2.2.3.6.1.2.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A3.T6.2.2.3.6.1.2.1.1" class="ltx_text ltx_font_bold">Theory</span></td>
</tr>
</table>
</td>
<td id="A3.T6.2.2.3.7" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;">
<table id="A3.T6.2.2.3.7.1" class="ltx_tabular ltx_align_middle">
<tr id="A3.T6.2.2.3.7.1.1" class="ltx_tr">
<td id="A3.T6.2.2.3.7.1.1.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="A3.T6.2.2.3.7.1.1.1.1" class="ltx_text ltx_font_bold">Counting</span> &amp;</td>
</tr>
<tr id="A3.T6.2.2.3.7.1.2" class="ltx_tr">
<td id="A3.T6.2.2.3.7.1.2.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A3.T6.2.2.3.7.1.2.1.1" class="ltx_text ltx_font_bold">Probability</span></td>
</tr>
</table>
</td>
<td id="A3.T6.2.2.3.8" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A3.T6.2.2.3.8.1" class="ltx_text ltx_font_bold">Prealgebra</span></td>
<td id="A3.T6.2.2.3.9" class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A3.T6.2.2.3.9.1" class="ltx_text ltx_font_bold">Algebra</span></td>
<td id="A3.T6.2.2.3.10" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A3.T6.2.2.3.10.1" class="ltx_text ltx_font_bold">Overall</span></td>
</tr>
<tr id="A3.T6.2.2.4" class="ltx_tr">
<td id="A3.T6.2.2.4.1" class="ltx_td ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"></td>
<td id="A3.T6.2.2.4.2" class="ltx_td ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"></td>
<td id="A3.T6.2.2.4.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;" colspan="8"><span id="A3.T6.2.2.4.3.1" class="ltx_text ltx_font_bold">Test Set</span></td>
</tr>
<tr id="A3.T6.1.1.1" class="ltx_tr">
<td id="A3.T6.1.1.1.1" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">ChatGPT (PAL) <span id="A3.T6.1.1.1.1.1" class="ltx_text" style="position:relative; bottom:-2.0pt;"><img src="/html/2309.17452/assets/figure/python.png" id="A3.T6.1.1.1.1.1.g1" class="ltx_graphics ltx_img_square" width="13" height="13" alt="[Uncaptioned image]"></span>
</td>
<td id="A3.T6.1.1.1.2" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A3.T6.1.1.1.2.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="A3.T6.1.1.1.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">18.5</td>
<td id="A3.T6.1.1.1.4" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">19.2</td>
<td id="A3.T6.1.1.1.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">23.2</td>
<td id="A3.T6.1.1.1.6" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">48.5</td>
<td id="A3.T6.1.1.1.7" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">43.0</td>
<td id="A3.T6.1.1.1.8" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">62.7</td>
<td id="A3.T6.1.1.1.9" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">45.4</td>
<td id="A3.T6.1.1.1.10" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">38.7</td>
</tr>
<tr id="A3.T6.2.2.2" class="ltx_tr">
<td id="A3.T6.2.2.2.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">GPT-4 (PAL) <span id="A3.T6.2.2.2.1.1" class="ltx_text" style="position:relative; bottom:-2.0pt;"><img src="/html/2309.17452/assets/figure/python.png" id="A3.T6.2.2.2.1.1.g1" class="ltx_graphics ltx_img_square" width="13" height="13" alt="[Uncaptioned image]"></span>
</td>
<td id="A3.T6.2.2.2.2" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A3.T6.2.2.2.2.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="A3.T6.2.2.2.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">32.8</td>
<td id="A3.T6.2.2.2.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">29.3</td>
<td id="A3.T6.2.2.2.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">38.0</td>
<td id="A3.T6.2.2.2.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">58.7</td>
<td id="A3.T6.2.2.2.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">61.0</td>
<td id="A3.T6.2.2.2.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">73.9</td>
<td id="A3.T6.2.2.2.9" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">59.1</td>
<td id="A3.T6.2.2.2.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">51.8</td>
</tr>
<tr id="A3.T6.2.2.5" class="ltx_tr">
<td id="A3.T6.2.2.5.1" class="ltx_td ltx_align_left" style="padding-left:4.0pt;padding-right:4.0pt;">GPT-4 (Tool-integrated Reasoning)</td>
<td id="A3.T6.2.2.5.2" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A3.T6.2.2.5.2.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="A3.T6.2.2.5.3" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">40.0</td>
<td id="A3.T6.2.2.5.4" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">37.2</td>
<td id="A3.T6.2.2.5.5" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">44.1</td>
<td id="A3.T6.2.2.5.6" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">68.9</td>
<td id="A3.T6.2.2.5.7" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">67.3</td>
<td id="A3.T6.2.2.5.8" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">82.2</td>
<td id="A3.T6.2.2.5.9" class="ltx_td ltx_align_center ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">75.8</td>
<td id="A3.T6.2.2.5.10" class="ltx_td ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">61.6</td>
</tr>
<tr id="A3.T6.2.2.6" class="ltx_tr">
<td id="A3.T6.2.2.6.1" class="ltx_td ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"></td>
<td id="A3.T6.2.2.6.2" class="ltx_td ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"></td>
<td id="A3.T6.2.2.6.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;" colspan="8"><span id="A3.T6.2.2.6.3.1" class="ltx_text ltx_font_bold">Training Set</span></td>
</tr>
<tr id="A3.T6.2.2.7" class="ltx_tr">
<td id="A3.T6.2.2.7.1" class="ltx_td ltx_align_left ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">GPT-4 (Tool-integrated Reasoning)</td>
<td id="A3.T6.2.2.7.2" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A3.T6.2.2.7.2.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="A3.T6.2.2.7.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">51.0</td>
<td id="A3.T6.2.2.7.4" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">51.5</td>
<td id="A3.T6.2.2.7.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">42.5</td>
<td id="A3.T6.2.2.7.6" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">77.4</td>
<td id="A3.T6.2.2.7.7" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">72.2</td>
<td id="A3.T6.2.2.7.8" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">89.8</td>
<td id="A3.T6.2.2.7.9" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">85.1</td>
<td id="A3.T6.2.2.7.10" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">64.3</td>
</tr>
<tr id="A3.T6.2.2.8" class="ltx_tr">
<td id="A3.T6.2.2.8.1" class="ltx_td ltx_align_left ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">w/ best@10</td>
<td id="A3.T6.2.2.8.2" class="ltx_td ltx_align_center ltx_border_bb ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A3.T6.2.2.8.2.1" class="ltx_text" style="color:#326400;">✓</span></td>
<td id="A3.T6.2.2.8.3" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">72.9</td>
<td id="A3.T6.2.2.8.4" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">70.0</td>
<td id="A3.T6.2.2.8.5" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">58.9</td>
<td id="A3.T6.2.2.8.6" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">91.6</td>
<td id="A3.T6.2.2.8.7" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">81.7</td>
<td id="A3.T6.2.2.8.8" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">95.5</td>
<td id="A3.T6.2.2.8.9" class="ltx_td ltx_align_center ltx_border_bb ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">96.3</td>
<td id="A3.T6.2.2.8.10" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">83.1</td>
</tr>
</table>
</span></div>
</figure>
<div id="A3.SS1.p1" class="ltx_para ltx_noindent">
<p id="A3.SS1.p1.1" class="ltx_p">Table <a href="#A3.T6" title="Table 6 ‣ C.1 Accuracies of Closed-Source Models on MATH ‣ Appendix C Additional Experiments and Analysis ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">6</span></a> presents the detailed accuracies of GPT-4 on the MATH dataset.
The Tool-integrated Reasoning method used by <span id="A3.SS1.p1.1.1" class="ltx_text ltx_font_smallcaps">ToRA</span> significantly outperforms PAL prompting when directly applied to the closed-source GPT-4, further demonstrating the benefits of synergizing natural language reasoning and program-based tool use.</p>
</div>
</section>
<section id="A3.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">C.2 </span>Effects of # Valid Trajectories for Output Space Shaping</h3>

<figure id="A3.F7" class="ltx_figure"><img src="/html/2309.17452/assets/x7.png" id="A3.F7.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="422" height="169" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 7: </span>
Effects of using different numbers of additional valid trajectories per question for output space shaping. </figcaption>
</figure>
<div id="A3.SS2.p1" class="ltx_para ltx_noindent">
<p id="A3.SS2.p1.1" class="ltx_p">As shown in Fig <a href="#A3.F7" title="Figure 7 ‣ C.2 Effects of # Valid Trajectories for Output Space Shaping ‣ Appendix C Additional Experiments and Analysis ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">7</span></a>, it is beneficial to increase the number of additional valid trajectories for output space shaping.</p>
</div>
</section>
<section id="A3.SS3" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">C.3 </span>Impact of Output Space Shaping in Relation to Question Difficulty</h3>

<figure id="A3.F8" class="ltx_figure"><img src="/html/2309.17452/assets/x8.png" id="A3.F8.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="422" height="282" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 8: </span>
Impact of Output Space Shaping in Relation to Question Difficulty.
</figcaption>
</figure>
<div id="A3.SS3.p1" class="ltx_para ltx_noindent">
<p id="A3.SS3.p1.1" class="ltx_p">We compare the effects of output space shaping on MATH problems of different difficulty levels (from level 1 to level 5) in Figure <a href="#A3.F8" title="Figure 8 ‣ C.3 Impact of Output Space Shaping in Relation to Question Difficulty ‣ Appendix C Additional Experiments and Analysis ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">8</span></a>, and present the statistics of MATH problems at different levels in Table <a href="#A3.T7" title="Table 7 ‣ C.3 Impact of Output Space Shaping in Relation to Question Difficulty ‣ Appendix C Additional Experiments and Analysis ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">7</span></a>. As can be seen:</p>
</div>
<div id="A3.SS3.p2" class="ltx_para ltx_noindent">
<ul id="A3.I1" class="ltx_itemize">
<li id="A3.I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="A3.I1.i1.p1" class="ltx_para">
<p id="A3.I1.i1.p1.1" class="ltx_p">Across these different difficulty levels and model sizes, output space shaping generally brings a significant improvement of 4.0% on average across different model sizes.</p>
</div>
</li>
<li id="A3.I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="A3.I1.i2.p1" class="ltx_para">
<p id="A3.I1.i2.p1.1" class="ltx_p">Output space shaping brings significant improvements for difficult, long problems. E.g., with <span id="A3.I1.i2.p1.1.1" class="ltx_text ltx_font_smallcaps">ToRA-Code</span>-13B, shaping does not significantly improve level 1 to level 2 problems, but it brings a substantial improvement of 5.4% to 5.7% for level 3 to level 5 problems.</p>
</div>
</li>
<li id="A3.I1.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="A3.I1.i3.p1" class="ltx_para ltx_noindent">
<p id="A3.I1.i3.p1.1" class="ltx_p">After using shaping, <span id="A3.I1.i3.p1.1.1" class="ltx_text ltx_font_smallcaps">ToRA-Code</span>-34B outperforms GPT-4 PAL on problems from Level 1 to Level 4, but there is still a gap at Level 5 (27.3% vs. 30.0%). These problems are usually longer (average about 248.4 characters), require more reasoning steps (&gt;1,000 characters) to solve, and more often include diagram inputs (about 20%). These observations may guide future work to focus more on solving these more difficult problems.</p>
</div>
</li>
</ul>
</div>
<figure id="A3.T7" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 7: </span>
Statistics of MATH problems at different levels. Average Answer Length indicates the average length of <span id="A3.T7.3.1" class="ltx_text ltx_font_smallcaps">ToRA</span> outputs; Training query coverage indicates the proportion of queries with at least one valid trajectory in <span id="A3.T7.4.2" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span> relative to the total queries in the original dataset.
</figcaption>
<table id="A3.T7.5" class="ltx_tabular ltx_centering ltx_align_middle">
<tr id="A3.T7.5.1" class="ltx_tr">
<td id="A3.T7.5.1.1" class="ltx_td ltx_border_tt"></td>
<td id="A3.T7.5.1.2" class="ltx_td ltx_align_center ltx_border_tt">Level 1</td>
<td id="A3.T7.5.1.3" class="ltx_td ltx_align_center ltx_border_tt">Level 2</td>
<td id="A3.T7.5.1.4" class="ltx_td ltx_align_center ltx_border_tt">Level 3</td>
<td id="A3.T7.5.1.5" class="ltx_td ltx_align_center ltx_border_tt">Level 4</td>
<td id="A3.T7.5.1.6" class="ltx_td ltx_align_center ltx_border_tt">Level 5</td>
</tr>
<tr id="A3.T7.5.2" class="ltx_tr">
<td id="A3.T7.5.2.1" class="ltx_td ltx_align_left ltx_border_t"># Test Samples</td>
<td id="A3.T7.5.2.2" class="ltx_td ltx_align_center ltx_border_t">437</td>
<td id="A3.T7.5.2.3" class="ltx_td ltx_align_center ltx_border_t">894</td>
<td id="A3.T7.5.2.4" class="ltx_td ltx_align_center ltx_border_t">1131</td>
<td id="A3.T7.5.2.5" class="ltx_td ltx_align_center ltx_border_t">1214</td>
<td id="A3.T7.5.2.6" class="ltx_td ltx_align_center ltx_border_t">1324</td>
</tr>
<tr id="A3.T7.5.3" class="ltx_tr">
<td id="A3.T7.5.3.1" class="ltx_td ltx_align_left">Avg Question Length</td>
<td id="A3.T7.5.3.2" class="ltx_td ltx_align_center">123.8</td>
<td id="A3.T7.5.3.3" class="ltx_td ltx_align_center">150.9</td>
<td id="A3.T7.5.3.4" class="ltx_td ltx_align_center">169.1</td>
<td id="A3.T7.5.3.5" class="ltx_td ltx_align_center">203.0</td>
<td id="A3.T7.5.3.6" class="ltx_td ltx_align_center">248.4</td>
</tr>
<tr id="A3.T7.5.4" class="ltx_tr">
<td id="A3.T7.5.4.1" class="ltx_td ltx_align_left">Avg Answer Length</td>
<td id="A3.T7.5.4.2" class="ltx_td ltx_align_center">503.1</td>
<td id="A3.T7.5.4.3" class="ltx_td ltx_align_center">655.8</td>
<td id="A3.T7.5.4.4" class="ltx_td ltx_align_center">751.2</td>
<td id="A3.T7.5.4.5" class="ltx_td ltx_align_center">881.6</td>
<td id="A3.T7.5.4.6" class="ltx_td ltx_align_center">1083.8</td>
</tr>
<tr id="A3.T7.5.5" class="ltx_tr">
<td id="A3.T7.5.5.1" class="ltx_td ltx_align_left ltx_border_bb">Training query coverage</td>
<td id="A3.T7.5.5.2" class="ltx_td ltx_align_center ltx_border_bb">97.7%</td>
<td id="A3.T7.5.5.3" class="ltx_td ltx_align_center ltx_border_bb">91.6%</td>
<td id="A3.T7.5.5.4" class="ltx_td ltx_align_center ltx_border_bb">86.5%</td>
<td id="A3.T7.5.5.5" class="ltx_td ltx_align_center ltx_border_bb">81.3%</td>
<td id="A3.T7.5.5.6" class="ltx_td ltx_align_center ltx_border_bb">68.0%</td>
</tr>
</table>
</figure>
</section>
</section>
<section id="A4" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix D </span>Detailed Information of <span id="A4.1.1" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span>
</h2>

<div id="A4.p1" class="ltx_para ltx_noindent">
<p id="A4.p1.1" class="ltx_p">We provide a more detailed introduction to the data construction process, quality control, and data statistical information, beyond Sec. <a href="#S2.SS2" title="2.2 Collecting Interactive Tool-Use Trajectories ‣ 2 ToRA: Tool-Integrated Agents for Mathematical Reasoning ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2.2</span></a>.</p>
</div>
<figure id="A4.T8" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 8: </span>Accuracy of <span id="A4.T8.3.1" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span> on GSM8k and MATH training set.
<span id="A4.T8.4.2" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span>-Greedy uses only the greedy trajectories, while ToRA-Corpus-16k combines sampled trajectories.</figcaption>
<div id="A4.T8.5" class="ltx_inline-block ltx_align_center ltx_transformed_outer" style="width:433.6pt;height:75.4pt;vertical-align:-0.8pt;"><span class="ltx_transformed_inner" style="transform:translate(-60.4pt,10.4pt) scale(0.78217684951155,0.78217684951155) ;">
<table id="A4.T8.5.1" class="ltx_tabular ltx_align_middle">
<tr id="A4.T8.5.1.1" class="ltx_tr">
<td id="A4.T8.5.1.1.1" class="ltx_td ltx_border_r ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"></td>
<td id="A4.T8.5.1.1.2" class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A4.T8.5.1.1.2.1" class="ltx_text ltx_font_bold">GSM8k</span></td>
<td id="A4.T8.5.1.1.3" class="ltx_td ltx_align_center ltx_border_tt" style="padding-left:4.0pt;padding-right:4.0pt;" colspan="8"><span id="A4.T8.5.1.1.3.1" class="ltx_text ltx_font_bold">MATH</span></td>
</tr>
<tr id="A4.T8.5.1.2" class="ltx_tr">
<td id="A4.T8.5.1.2.1" class="ltx_td ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;"></td>
<td id="A4.T8.5.1.2.2" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A4.T8.5.1.2.2.1" class="ltx_text ltx_font_bold">All</span></td>
<td id="A4.T8.5.1.2.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A4.T8.5.1.2.3.1" class="ltx_text ltx_font_bold">All</span></td>
<td id="A4.T8.5.1.2.4" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">
<table id="A4.T8.5.1.2.4.1" class="ltx_tabular ltx_align_middle">
<tr id="A4.T8.5.1.2.4.1.1" class="ltx_tr">
<td id="A4.T8.5.1.2.4.1.1.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A4.T8.5.1.2.4.1.1.1.1" class="ltx_text ltx_font_bold">Intermediate</span></td>
</tr>
<tr id="A4.T8.5.1.2.4.1.2" class="ltx_tr">
<td id="A4.T8.5.1.2.4.1.2.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A4.T8.5.1.2.4.1.2.1.1" class="ltx_text ltx_font_bold">Algebra</span></td>
</tr>
</table>
</td>
<td id="A4.T8.5.1.2.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A4.T8.5.1.2.5.1" class="ltx_text ltx_font_bold">Precalculus</span></td>
<td id="A4.T8.5.1.2.6" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A4.T8.5.1.2.6.1" class="ltx_text ltx_font_bold">Geometry</span></td>
<td id="A4.T8.5.1.2.7" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">
<table id="A4.T8.5.1.2.7.1" class="ltx_tabular ltx_align_middle">
<tr id="A4.T8.5.1.2.7.1.1" class="ltx_tr">
<td id="A4.T8.5.1.2.7.1.1.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A4.T8.5.1.2.7.1.1.1.1" class="ltx_text ltx_font_bold">Number</span></td>
</tr>
<tr id="A4.T8.5.1.2.7.1.2" class="ltx_tr">
<td id="A4.T8.5.1.2.7.1.2.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A4.T8.5.1.2.7.1.2.1.1" class="ltx_text ltx_font_bold">Theory</span></td>
</tr>
</table>
</td>
<td id="A4.T8.5.1.2.8" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">
<table id="A4.T8.5.1.2.8.1" class="ltx_tabular ltx_align_middle">
<tr id="A4.T8.5.1.2.8.1.1" class="ltx_tr">
<td id="A4.T8.5.1.2.8.1.1.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="A4.T8.5.1.2.8.1.1.1.1" class="ltx_text ltx_font_bold">Counting</span> &amp;</td>
</tr>
<tr id="A4.T8.5.1.2.8.1.2" class="ltx_tr">
<td id="A4.T8.5.1.2.8.1.2.1" class="ltx_td ltx_nopad_r ltx_align_center" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A4.T8.5.1.2.8.1.2.1.1" class="ltx_text ltx_font_bold">Probability</span></td>
</tr>
</table>
</td>
<td id="A4.T8.5.1.2.9" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A4.T8.5.1.2.9.1" class="ltx_text ltx_font_bold">Prealgebra</span></td>
<td id="A4.T8.5.1.2.10" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;"><span id="A4.T8.5.1.2.10.1" class="ltx_text ltx_font_bold">Algebra</span></td>
</tr>
<tr id="A4.T8.5.1.3" class="ltx_tr">
<td id="A4.T8.5.1.3.1" class="ltx_td ltx_align_left ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="A4.T8.5.1.3.1.1" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span>-Greedy</td>
<td id="A4.T8.5.1.3.2" class="ltx_td ltx_align_center ltx_border_r ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">94.4</td>
<td id="A4.T8.5.1.3.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">64.3</td>
<td id="A4.T8.5.1.3.4" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">51.0</td>
<td id="A4.T8.5.1.3.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">51.5</td>
<td id="A4.T8.5.1.3.6" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">70.0</td>
<td id="A4.T8.5.1.3.7" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">77.4</td>
<td id="A4.T8.5.1.3.8" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">72.2</td>
<td id="A4.T8.5.1.3.9" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">89.8</td>
<td id="A4.T8.5.1.3.10" class="ltx_td ltx_align_center ltx_border_t" style="padding-left:4.0pt;padding-right:4.0pt;">85.1</td>
</tr>
<tr id="A4.T8.5.1.4" class="ltx_tr">
<td id="A4.T8.5.1.4.1" class="ltx_td ltx_align_left ltx_border_bb ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">
<span id="A4.T8.5.1.4.1.1" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span>-16k</td>
<td id="A4.T8.5.1.4.2" class="ltx_td ltx_align_center ltx_border_bb ltx_border_r" style="padding-left:4.0pt;padding-right:4.0pt;">98.2</td>
<td id="A4.T8.5.1.4.3" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">83.1</td>
<td id="A4.T8.5.1.4.4" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">72.9</td>
<td id="A4.T8.5.1.4.5" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">70.0</td>
<td id="A4.T8.5.1.4.6" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">58.9</td>
<td id="A4.T8.5.1.4.7" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">91.6</td>
<td id="A4.T8.5.1.4.8" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">81.7</td>
<td id="A4.T8.5.1.4.9" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">95.5</td>
<td id="A4.T8.5.1.4.10" class="ltx_td ltx_align_center ltx_border_bb" style="padding-left:4.0pt;padding-right:4.0pt;">96.3</td>
</tr>
</table>
</span></div>
</figure>
<figure id="A4.T9" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 9: </span>Statistics of <span id="A4.T9.2.1" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span>-16k</figcaption>
<table id="A4.T9.3" class="ltx_tabular ltx_centering ltx_align_middle">
<tr id="A4.T9.3.1" class="ltx_tr">
<td id="A4.T9.3.1.1" class="ltx_td ltx_border_tt"></td>
<td id="A4.T9.3.1.2" class="ltx_td ltx_align_right ltx_border_tt"><span id="A4.T9.3.1.2.1" class="ltx_text ltx_font_bold">GSM8k</span></td>
<td id="A4.T9.3.1.3" class="ltx_td ltx_align_right ltx_border_tt"><span id="A4.T9.3.1.3.1" class="ltx_text ltx_font_bold">MATH</span></td>
<td id="A4.T9.3.1.4" class="ltx_td ltx_align_right ltx_border_tt"><span id="A4.T9.3.1.4.1" class="ltx_text ltx_font_bold">Total</span></td>
</tr>
<tr id="A4.T9.3.2" class="ltx_tr">
<td id="A4.T9.3.2.1" class="ltx_td ltx_align_left ltx_border_t"># Train Samples</td>
<td id="A4.T9.3.2.2" class="ltx_td ltx_align_right ltx_border_t">7,657</td>
<td id="A4.T9.3.2.3" class="ltx_td ltx_align_right ltx_border_t">7,881</td>
<td id="A4.T9.3.2.4" class="ltx_td ltx_align_right ltx_border_t">15,538</td>
</tr>
<tr id="A4.T9.3.3" class="ltx_tr">
<td id="A4.T9.3.3.1" class="ltx_td ltx_align_left">Avg Question Length</td>
<td id="A4.T9.3.3.2" class="ltx_td ltx_align_right">236</td>
<td id="A4.T9.3.3.3" class="ltx_td ltx_align_right">189</td>
<td id="A4.T9.3.3.4" class="ltx_td ltx_align_right">211</td>
</tr>
<tr id="A4.T9.3.4" class="ltx_tr">
<td id="A4.T9.3.4.1" class="ltx_td ltx_align_left">Avg Trajectory Length</td>
<td id="A4.T9.3.4.2" class="ltx_td ltx_align_right">678</td>
<td id="A4.T9.3.4.3" class="ltx_td ltx_align_right">704</td>
<td id="A4.T9.3.4.4" class="ltx_td ltx_align_right">691</td>
</tr>
<tr id="A4.T9.3.5" class="ltx_tr">
<td id="A4.T9.3.5.1" class="ltx_td ltx_align_left">Min Trajectory Length</td>
<td id="A4.T9.3.5.2" class="ltx_td ltx_align_right">218</td>
<td id="A4.T9.3.5.3" class="ltx_td ltx_align_right">119</td>
<td id="A4.T9.3.5.4" class="ltx_td ltx_align_right">119</td>
</tr>
<tr id="A4.T9.3.6" class="ltx_tr">
<td id="A4.T9.3.6.1" class="ltx_td ltx_align_left ltx_border_bb">Max Trajectory Length</td>
<td id="A4.T9.3.6.2" class="ltx_td ltx_align_right ltx_border_bb">1,713</td>
<td id="A4.T9.3.6.3" class="ltx_td ltx_align_right ltx_border_bb">2,486</td>
<td id="A4.T9.3.6.4" class="ltx_td ltx_align_right ltx_border_bb">2,486</td>
</tr>
</table>
</figure>
<section id="A4.SS0.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Data Format and Quality Control</h5>

<div id="A4.SS0.SSS0.Px1.p1" class="ltx_para ltx_noindent">
<p id="A4.SS0.SSS0.Px1.p1.1" class="ltx_p">In our preliminary experiments, we found that the tool-integrated reasoning trajectory format generated by zero-shot prompting was somewhat chaotic. Therefore, we designed a few-shot prompting to control the reasoning format, which effectively improved data quality. On the other hand, we increased the annotation success rate by sampling, ensuring more comprehensive coverage of the training query.</p>
</div>
</section>
<section id="A4.SS0.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Data Filtering Process</h5>

<div id="A4.SS0.SSS0.Px2.p1" class="ltx_para ltx_noindent">
<p id="A4.SS0.SSS0.Px2.p1.1" class="ltx_p">For the data constructed, we filtered out paths that produced incorrect answers by matching them with standard answers. To prevent the model from learning incorrect intermediate reasoning processes, we further filtered out data samples with intermediate program execution errors.</p>
</div>
</section>
<section id="A4.SS0.SSS0.Px3" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Dataset Statistics</h5>

<div id="A4.SS0.SSS0.Px3.p1" class="ltx_para ltx_noindent">
<p id="A4.SS0.SSS0.Px3.p1.1" class="ltx_p">In Table <a href="#A4.T8" title="Table 8 ‣ Appendix D Detailed Information of ToRA-Corpus ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">8</span></a>, we compared the annotation accuracy (i.e., sample coverage) of the training set on GSM8k, MATH, and MATH subtopics of <span id="A4.SS0.SSS0.Px3.p1.1.1" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span>-Greedy using only the greedy trajectories, and <span id="A4.SS0.SSS0.Px3.p1.1.2" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span>-16k combined with sampled trajectories. Furthermore, in Table <a href="#A4.T9" title="Table 9 ‣ Appendix D Detailed Information of ToRA-Corpus ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">9</span></a>, we reported the statistical data of <span id="A4.SS0.SSS0.Px3.p1.1.3" class="ltx_text ltx_font_smallcaps">ToRA-Corpus</span>-16k, such as the number of samples, average question length, average, minimum, and maximum trajectory length, as shown in the following tables.</p>
</div>
</section>
<section id="A4.SS0.SSS0.Px4" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Rationale as Hints</h5>

<div id="A4.SS0.SSS0.Px4.p1" class="ltx_para ltx_noindent">
<p id="A4.SS0.SSS0.Px4.p1.1" class="ltx_p">As described in Section <a href="#S2.SS2" title="2.2 Collecting Interactive Tool-Use Trajectories ‣ 2 ToRA: Tool-Integrated Agents for Mathematical Reasoning ‣ \logo ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2.2</span></a>, we annotated interactive tool-use trajectories for the training questions from MATH with GPT-4.
GPT-4 achieves a success rate below 65% using greedy decoding.
As MATH was originally annotated with natural language rationales, to improve the annotation success rate, we tried to provide GPT-4 with the human rationales as hints <cite class="ltx_cite ltx_citemacro_citep">(Zelikman et al., <a href="#bib.bib49" title="" class="ltx_ref">2022</a>)</cite>.
However, when using this method, GPT-4 tends to replicate the hints and ignore tool-use outputs especially when the outputs are inconsistent with the hints, thus failing to produce high-quality trajectories.
Hence, we deferred the utilization of the already-annotated natural language rationales for future investigations.
Instead, we employed nucleus sampling to recall valid trajectories for questions that remained unsolved through greedy decoding.
This approach significantly boosted annotation accuracy to 83.1%.</p>
</div>
<div class="ltx_pagination ltx_role_newpage"></div>
</section>
</section>
<section id="A5" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix E </span>Prompts</h2>

<div id="A5.p1" class="ltx_para ltx_noindent">
<p id="A5.p1.1" class="ltx_p">We present instructions and example few-shot prompts of Tool-integrated Reasoning for querying GPT-4.</p>
</div>
<figure id="LST1" class="ltx_float ltx_lstlisting">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float">Listing 1: </span>Prompts of Tool-integrated Reasoning for GPT-4.</figcaption>
<div id="LST1.1" class="ltx_listing ltx_lstlisting ltx_framed ltx_framed_topbottom ltx_listing">
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<div id="lstnumberx1" class="ltx_listingline">
<span id="lstnumberx1.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Integrate</span><span id="lstnumberx1.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx1.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">step</span><span id="lstnumberx1.4" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx1.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">by</span><span id="lstnumberx1.6" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx1.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">step</span><span id="lstnumberx1.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx1.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">reasoning</span><span id="lstnumberx1.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx1.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx1.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx1.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Python</span><span id="lstnumberx1.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx1.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">code</span><span id="lstnumberx1.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx1.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx1.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx1.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve</span><span id="lstnumberx1.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx1.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">math</span><span id="lstnumberx1.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx1.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">problems</span><span id="lstnumberx1.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx1.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">using</span><span id="lstnumberx1.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx1.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx1.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx1.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">following</span><span id="lstnumberx1.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx1.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">guidelines</span><span id="lstnumberx1.32" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx2" class="ltx_listingline">
</div>
<div id="lstnumberx3" class="ltx_listingline">
<span id="lstnumberx3.1" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx3.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx3.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Analyze</span><span id="lstnumberx3.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx3.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx3.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx3.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">question</span><span id="lstnumberx3.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx3.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx3.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx3.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">write</span><span id="lstnumberx3.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx3.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">functions</span><span id="lstnumberx3.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx3.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx3.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx3.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve</span><span id="lstnumberx3.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx3.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx3.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx3.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">problem</span><span id="lstnumberx3.22" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx3.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx3.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx3.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx3.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">function</span><span id="lstnumberx3.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx3.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">should</span><span id="lstnumberx3.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx3.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">not</span><span id="lstnumberx3.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx3.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">take</span><span id="lstnumberx3.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx3.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">any</span><span id="lstnumberx3.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx3.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">arguments</span><span id="lstnumberx3.37" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx4" class="ltx_listingline">
<span id="lstnumberx4.1" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx4.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx4.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Present</span><span id="lstnumberx4.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx4.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx4.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx4.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">final</span><span id="lstnumberx4.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx4.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx4.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx4.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx4.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx4.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">LaTeX</span><span id="lstnumberx4.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx4.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">using</span><span id="lstnumberx4.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx4.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx4.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx4.19" class="ltx_text ltx_font_typewriter">‘\</span><span id="lstnumberx4.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx4.21" class="ltx_text ltx_font_typewriter">{}‘</span><span id="lstnumberx4.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx4.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">without</span><span id="lstnumberx4.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx4.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">any</span><span id="lstnumberx4.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx4.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">units</span><span id="lstnumberx4.28" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx5" class="ltx_listingline">
<span id="lstnumberx5.1" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx5.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Utilize</span><span id="lstnumberx5.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx5.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.7" class="ltx_text ltx_font_typewriter">‘</span><span id="lstnumberx5.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pi</span><span id="lstnumberx5.9" class="ltx_text ltx_font_typewriter">‘</span><span id="lstnumberx5.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbol</span><span id="lstnumberx5.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx5.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.15" class="ltx_text ltx_font_typewriter">‘</span><span id="lstnumberx5.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Rational</span><span id="lstnumberx5.17" class="ltx_text ltx_font_typewriter">‘‘</span><span id="lstnumberx5.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx5.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Sympy</span><span id="lstnumberx5.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx5.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx5.26" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx5.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pi$</span><span id="lstnumberx5.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx5.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">fractions</span><span id="lstnumberx5.32" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx5.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx5.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">simplify</span><span id="lstnumberx5.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">all</span><span id="lstnumberx5.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">fractions</span><span id="lstnumberx5.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx5.43" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">square</span><span id="lstnumberx5.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots</span><span id="lstnumberx5.47" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.48" class="ltx_text ltx_lst_identifier ltx_font_typewriter">without</span><span id="lstnumberx5.49" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.50" class="ltx_text ltx_lst_identifier ltx_font_typewriter">converting</span><span id="lstnumberx5.51" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">them</span><span id="lstnumberx5.53" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.54" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx5.55" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.56" class="ltx_text ltx_lst_identifier ltx_font_typewriter">decimal</span><span id="lstnumberx5.57" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx5.58" class="ltx_text ltx_lst_identifier ltx_font_typewriter">values</span><span id="lstnumberx5.59" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx6" class="ltx_listingline">
</div>
<div id="lstnumberx7" class="ltx_listingline">
<span id="lstnumberx7.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Here</span><span id="lstnumberx7.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx7.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx7.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx7.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">some</span><span id="lstnumberx7.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx7.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">examples</span><span id="lstnumberx7.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx7.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">you</span><span id="lstnumberx7.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx7.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">may</span><span id="lstnumberx7.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx7.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">refer</span><span id="lstnumberx7.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx7.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx7.16" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx8" class="ltx_listingline">
</div>
<div id="lstnumberx9" class="ltx_listingline">
<span id="lstnumberx9.1" class="ltx_text ltx_font_typewriter">---</span>
</div>
<div id="lstnumberx10" class="ltx_listingline">
</div>
<div id="lstnumberx11" class="ltx_listingline">
<span id="lstnumberx11.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Question</span><span id="lstnumberx11.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx11.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Convert</span><span id="lstnumberx11.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx11.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">point</span><span id="lstnumberx11.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx11.11" class="ltx_text ltx_font_typewriter">(0,</span><span id="lstnumberx11.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.13" class="ltx_text ltx_font_typewriter">-3</span><span id="lstnumberx11.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.15" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx11.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx11.17" class="ltx_text ltx_font_typewriter">{3},</span><span id="lstnumberx11.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.19" class="ltx_text ltx_font_typewriter">3)</span><span id="lstnumberx11.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx11.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx11.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rectangular</span><span id="lstnumberx11.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">coordinates</span><span id="lstnumberx11.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx11.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">spherical</span><span id="lstnumberx11.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">coordinates</span><span id="lstnumberx11.33" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx11.34" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx11.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Enter</span><span id="lstnumberx11.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">your</span><span id="lstnumberx11.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">answer</span><span id="lstnumberx11.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx11.42" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx11.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">form</span><span id="lstnumberx11.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx11.48" class="ltx_text ltx_font_typewriter">(\</span><span id="lstnumberx11.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rho</span><span id="lstnumberx11.50" class="ltx_text ltx_font_typewriter">,\</span><span id="lstnumberx11.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">theta</span><span id="lstnumberx11.52" class="ltx_text ltx_font_typewriter">,\</span><span id="lstnumberx11.53" class="ltx_text ltx_lst_identifier ltx_font_typewriter">phi</span><span id="lstnumberx11.54" class="ltx_text ltx_font_typewriter">),</span><span id="lstnumberx11.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx11.56" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">where</span><span id="lstnumberx11.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx11.60" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx11.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rho</span><span id="lstnumberx11.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.63" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx11.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.65" class="ltx_text ltx_font_typewriter">0,</span><span id="lstnumberx11.66" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx11.67" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.68" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$0</span><span id="lstnumberx11.69" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.70" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx11.71" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx11.72" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.73" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx11.74" class="ltx_text ltx_lst_identifier ltx_font_typewriter">theta</span><span id="lstnumberx11.75" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.76" class="ltx_text ltx_font_typewriter">&lt;</span><span id="lstnumberx11.77" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.78" class="ltx_text ltx_font_typewriter">2</span><span id="lstnumberx11.79" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.80" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx11.81" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pi</span><span id="lstnumberx11.82" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx11.83" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx11.84" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.85" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx11.86" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.87" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$0</span><span id="lstnumberx11.88" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.89" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx11.90" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx11.91" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.92" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx11.93" class="ltx_text ltx_lst_identifier ltx_font_typewriter">phi</span><span id="lstnumberx11.94" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.95" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx11.96" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx11.97" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx11.98" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx11.99" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pi</span><span id="lstnumberx11.100" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx11.101" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span>
</div>
<div id="lstnumberx12" class="ltx_listingline">
</div>
<div id="lstnumberx13" class="ltx_listingline">
<span id="lstnumberx13.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solution</span><span id="lstnumberx13.2" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx14" class="ltx_listingline">
<span id="lstnumberx14.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">To</span><span id="lstnumberx14.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">convert</span><span id="lstnumberx14.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx14.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">point</span><span id="lstnumberx14.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx14.10" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx14.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx14.12" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx14.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx14.15" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx14.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx14.18" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx14.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx14.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx14.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rectangular</span><span id="lstnumberx14.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">coordinates</span><span id="lstnumberx14.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx14.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">spherical</span><span id="lstnumberx14.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">coordinates</span><span id="lstnumberx14.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx14.34" class="ltx_text ltx_font_typewriter">(\</span><span id="lstnumberx14.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rho</span><span id="lstnumberx14.36" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx14.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.38" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx14.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">theta</span><span id="lstnumberx14.40" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx14.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.42" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx14.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">phi</span><span id="lstnumberx14.44" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx14.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx14.46" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx14.47" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.48" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx14.49" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.50" class="ltx_text ltx_lst_identifier ltx_font_typewriter">use</span><span id="lstnumberx14.51" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx14.53" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.54" class="ltx_text ltx_lst_identifier ltx_font_typewriter">following</span><span id="lstnumberx14.55" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx14.56" class="ltx_text ltx_lst_identifier ltx_font_typewriter">formulas</span><span id="lstnumberx14.57" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx15" class="ltx_listingline">
<span id="lstnumberx15.1" class="ltx_text ltx_font_typewriter">1.</span><span id="lstnumberx15.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx15.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx15.4" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx15.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rho</span><span id="lstnumberx15.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx15.7" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx15.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx15.9" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx15.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx15.11" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx15.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx15.13" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx15.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx15.15" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx15.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx15.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx15.18" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx15.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx15.20" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx15.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx15.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx15.23" class="ltx_text ltx_font_typewriter">^2}</span><span id="lstnumberx15.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span>
</div>
<div id="lstnumberx16" class="ltx_listingline">
<span id="lstnumberx16.1" class="ltx_text ltx_font_typewriter">2.</span><span id="lstnumberx16.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx16.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx16.4" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx16.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">theta</span><span id="lstnumberx16.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx16.7" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx16.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx16.9" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx16.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">arctan</span><span id="lstnumberx16.11" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx16.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx16.13" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx16.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx16.15" class="ltx_text ltx_font_typewriter">}{</span><span id="lstnumberx16.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx16.17" class="ltx_text ltx_font_typewriter">}}</span><span id="lstnumberx16.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span>
</div>
<div id="lstnumberx17" class="ltx_listingline">
<span id="lstnumberx17.1" class="ltx_text ltx_font_typewriter">3.</span><span id="lstnumberx17.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx17.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx17.4" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx17.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">phi</span><span id="lstnumberx17.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx17.7" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx17.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx17.9" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx17.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">arccos</span><span id="lstnumberx17.11" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx17.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx17.13" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx17.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx17.15" class="ltx_text ltx_font_typewriter">}{\</span><span id="lstnumberx17.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rho</span><span id="lstnumberx17.17" class="ltx_text ltx_font_typewriter">}}</span><span id="lstnumberx17.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span>
</div>
<div id="lstnumberx18" class="ltx_listingline">
</div>
<div id="lstnumberx19" class="ltx_listingline">
<span id="lstnumberx19.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Let</span><span id="lstnumberx19.2" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx19.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx19.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx19.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">write</span><span id="lstnumberx19.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx19.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx19.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx19.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">function</span><span id="lstnumberx19.10" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx20" class="ltx_listingline">
</div>
<div id="lstnumberx21" class="ltx_listingline">
<span id="lstnumberx21.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx21.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span>
</div>
<div id="lstnumberx22" class="ltx_listingline">
<span id="lstnumberx22.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx22.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx22.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx22.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx22.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx22.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx22.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx22.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx22.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx22.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">atan2</span><span id="lstnumberx22.11" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx22.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx22.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">acos</span><span id="lstnumberx22.14" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx22.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx22.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pi</span>
</div>
<div id="lstnumberx23" class="ltx_listingline">
</div>
<div id="lstnumberx24" class="ltx_listingline">
<span id="lstnumberx24.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">def</span><span id="lstnumberx24.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx24.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rectangular_to_spherical</span><span id="lstnumberx24.4" class="ltx_text ltx_font_typewriter">():</span>
</div>
<div id="lstnumberx25" class="ltx_listingline">
<span id="lstnumberx25.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx25.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx25.3" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx25.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx25.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx25.6" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx25.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx25.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx25.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx25.10" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx25.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx25.12" class="ltx_text ltx_font_typewriter">0,</span><span id="lstnumberx25.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx25.14" class="ltx_text ltx_font_typewriter">-3*</span><span id="lstnumberx25.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx25.16" class="ltx_text ltx_font_typewriter">(3),</span><span id="lstnumberx25.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx25.18" class="ltx_text ltx_font_typewriter">3</span>
</div>
<div id="lstnumberx26" class="ltx_listingline">
<span id="lstnumberx26.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx26.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rho</span><span id="lstnumberx26.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx26.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx26.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx26.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx26.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx26.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx26.9" class="ltx_text ltx_font_typewriter">**2</span><span id="lstnumberx26.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx26.11" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx26.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx26.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx26.14" class="ltx_text ltx_font_typewriter">**2</span><span id="lstnumberx26.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx26.16" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx26.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx26.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx26.19" class="ltx_text ltx_font_typewriter">**2)</span>
</div>
<div id="lstnumberx27" class="ltx_listingline">
<span id="lstnumberx27.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx27.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">theta</span><span id="lstnumberx27.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx27.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx27.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx27.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">atan2</span><span id="lstnumberx27.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx27.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx27.9" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx27.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx27.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx27.12" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx28" class="ltx_listingline">
<span id="lstnumberx28.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx28.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">phi</span><span id="lstnumberx28.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx28.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx28.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx28.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">acos</span><span id="lstnumberx28.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx28.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx28.9" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx28.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rho</span><span id="lstnumberx28.11" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx29" class="ltx_listingline">
<span id="lstnumberx29.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx29.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx29.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx29.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rho</span><span id="lstnumberx29.5" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx29.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx29.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">theta</span><span id="lstnumberx29.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx29.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx29.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">phi</span>
</div>
<div id="lstnumberx30" class="ltx_listingline">
</div>
<div id="lstnumberx31" class="ltx_listingline">
<span id="lstnumberx31.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">spherical_coordinates</span><span id="lstnumberx31.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx31.3" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx31.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx31.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rectangular_to_spherical</span><span id="lstnumberx31.6" class="ltx_text ltx_font_typewriter">()</span>
</div>
<div id="lstnumberx32" class="ltx_listingline">
<span id="lstnumberx32.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">print</span><span id="lstnumberx32.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx32.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">spherical_coordinates</span><span id="lstnumberx32.4" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx33" class="ltx_listingline">
<span id="lstnumberx33.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx34" class="ltx_listingline">
<span id="lstnumberx34.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx34.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span>
</div>
<div id="lstnumberx35" class="ltx_listingline">
<span id="lstnumberx35.1" class="ltx_text ltx_font_typewriter">(6,</span><span id="lstnumberx35.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx35.3" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx35.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pi</span><span id="lstnumberx35.5" class="ltx_text ltx_font_typewriter">/2,</span><span id="lstnumberx35.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx35.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pi</span><span id="lstnumberx35.8" class="ltx_text ltx_font_typewriter">/3)</span>
</div>
<div id="lstnumberx36" class="ltx_listingline">
<span id="lstnumberx36.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx37" class="ltx_listingline">
</div>
<div id="lstnumberx38" class="ltx_listingline">
<span id="lstnumberx38.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx38.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">required</span><span id="lstnumberx38.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">form</span><span id="lstnumberx38.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx38.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx38.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">answer</span><span id="lstnumberx38.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx38.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx38.16" class="ltx_text ltx_font_typewriter">(\</span><span id="lstnumberx38.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rho</span><span id="lstnumberx38.18" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx38.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.20" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx38.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">theta</span><span id="lstnumberx38.22" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx38.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.24" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx38.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">phi</span><span id="lstnumberx38.26" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx38.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx38.28" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx38.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">where</span><span id="lstnumberx38.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx38.33" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx38.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rho</span><span id="lstnumberx38.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.36" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx38.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.38" class="ltx_text ltx_font_typewriter">0</span><span id="lstnumberx38.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx38.40" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx38.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$0</span><span id="lstnumberx38.43" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.44" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx38.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx38.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.47" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx38.48" class="ltx_text ltx_lst_identifier ltx_font_typewriter">theta</span><span id="lstnumberx38.49" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.50" class="ltx_text ltx_font_typewriter">&lt;</span><span id="lstnumberx38.51" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.52" class="ltx_text ltx_font_typewriter">2\</span><span id="lstnumberx38.53" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pi$</span><span id="lstnumberx38.54" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx38.55" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.56" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx38.57" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.58" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$0</span><span id="lstnumberx38.59" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.60" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx38.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx38.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.63" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx38.64" class="ltx_text ltx_lst_identifier ltx_font_typewriter">phi</span><span id="lstnumberx38.65" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.66" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx38.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx38.68" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.69" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx38.70" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pi$</span><span id="lstnumberx38.71" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx38.72" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.73" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Since</span><span id="lstnumberx38.74" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.75" class="ltx_text ltx_lst_identifier ltx_font_typewriter">our</span><span id="lstnumberx38.76" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.77" class="ltx_text ltx_lst_identifier ltx_font_typewriter">initial</span><span id="lstnumberx38.78" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.79" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span><span id="lstnumberx38.80" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.81" class="ltx_text ltx_lst_identifier ltx_font_typewriter">has</span><span id="lstnumberx38.82" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.83" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx38.84" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx38.85" class="ltx_text ltx_lst_identifier ltx_font_typewriter">theta</span><span id="lstnumberx38.86" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.87" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx38.88" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.89" class="ltx_text ltx_font_typewriter">-\</span><span id="lstnumberx38.90" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pi</span><span id="lstnumberx38.91" class="ltx_text ltx_font_typewriter">/2</span><span id="lstnumberx38.92" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx38.93" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx38.94" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.95" class="ltx_text ltx_lst_identifier ltx_font_typewriter">which</span><span id="lstnumberx38.96" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.97" class="ltx_text ltx_lst_identifier ltx_font_typewriter">doesn</span><span id="lstnumberx38.98" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx38.99" class="ltx_text ltx_lst_identifier ltx_font_typewriter">t</span><span id="lstnumberx38.100" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.101" class="ltx_text ltx_lst_identifier ltx_font_typewriter">meet</span><span id="lstnumberx38.102" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.103" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx38.104" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.105" class="ltx_text ltx_lst_identifier ltx_font_typewriter">condition</span><span id="lstnumberx38.106" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.107" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx38.108" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.109" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx38.110" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx38.111" class="ltx_text ltx_lst_identifier ltx_font_typewriter">theta$</span><span id="lstnumberx38.112" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx38.113" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.114" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx38.115" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.116" class="ltx_text ltx_lst_identifier ltx_font_typewriter">add</span><span id="lstnumberx38.117" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.118" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$2</span><span id="lstnumberx38.119" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx38.120" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pi$</span><span id="lstnumberx38.121" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.122" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx38.123" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.124" class="ltx_text ltx_lst_identifier ltx_font_typewriter">adjust</span><span id="lstnumberx38.125" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.126" class="ltx_text ltx_lst_identifier ltx_font_typewriter">its</span><span id="lstnumberx38.127" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx38.128" class="ltx_text ltx_lst_identifier ltx_font_typewriter">value</span><span id="lstnumberx38.129" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx39" class="ltx_listingline">
</div>
<div id="lstnumberx40" class="ltx_listingline">
<span id="lstnumberx40.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Hence</span><span id="lstnumberx40.2" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx40.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx40.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx40.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx40.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">spherical</span><span id="lstnumberx40.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx40.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">coordinates</span><span id="lstnumberx40.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx40.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx40.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx40.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx40.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx40.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">point</span><span id="lstnumberx40.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx40.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx40.17" class="ltx_text ltx_font_typewriter">(0,</span><span id="lstnumberx40.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx40.19" class="ltx_text ltx_font_typewriter">-3\</span><span id="lstnumberx40.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx40.21" class="ltx_text ltx_font_typewriter">{3},</span><span id="lstnumberx40.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx40.23" class="ltx_text ltx_font_typewriter">3)</span><span id="lstnumberx40.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx40.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx40.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx40.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx40.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx40.29" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx40.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx40.31" class="ltx_text ltx_font_typewriter">{(6,</span><span id="lstnumberx40.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx40.33" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx40.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx40.35" class="ltx_text ltx_font_typewriter">{3\</span><span id="lstnumberx40.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pi</span><span id="lstnumberx40.37" class="ltx_text ltx_font_typewriter">}{2},</span><span id="lstnumberx40.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx40.39" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx40.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx40.41" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx40.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pi</span><span id="lstnumberx40.43" class="ltx_text ltx_font_typewriter">}{3})}</span><span id="lstnumberx40.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx40.45" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx41" class="ltx_listingline">
</div>
<div id="lstnumberx42" class="ltx_listingline">
<span id="lstnumberx42.1" class="ltx_text ltx_font_typewriter">---</span>
</div>
<div id="lstnumberx43" class="ltx_listingline">
</div>
<div id="lstnumberx44" class="ltx_listingline">
<span id="lstnumberx44.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Question</span><span id="lstnumberx44.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx44.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx44.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Compute</span><span id="lstnumberx44.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx44.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$1011_2</span><span id="lstnumberx44.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx44.8" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx44.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx44.10" class="ltx_text ltx_font_typewriter">101</span><span id="lstnumberx44.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_2</span><span id="lstnumberx44.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx44.13" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx44.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx44.15" class="ltx_text ltx_font_typewriter">1100</span><span id="lstnumberx44.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_2</span><span id="lstnumberx44.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx44.18" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx44.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx44.20" class="ltx_text ltx_font_typewriter">1101</span><span id="lstnumberx44.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_2$</span><span id="lstnumberx44.22" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx44.23" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx44.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Express</span><span id="lstnumberx44.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx44.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">your</span><span id="lstnumberx44.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx44.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">answer</span><span id="lstnumberx44.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx44.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx44.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx44.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">base</span><span id="lstnumberx44.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx44.34" class="ltx_text ltx_font_typewriter">2.</span>
</div>
<div id="lstnumberx45" class="ltx_listingline">
</div>
<div id="lstnumberx46" class="ltx_listingline">
<span id="lstnumberx46.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solution</span><span id="lstnumberx46.2" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx47" class="ltx_listingline">
<span id="lstnumberx47.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Let</span><span id="lstnumberx47.2" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx47.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx47.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx47.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">write</span><span id="lstnumberx47.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx47.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx47.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx47.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">single</span><span id="lstnumberx47.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx47.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span><span id="lstnumberx47.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx47.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">function</span><span id="lstnumberx47.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx47.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx47.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx47.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">binary</span><span id="lstnumberx47.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx47.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">computation</span><span id="lstnumberx47.20" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx48" class="ltx_listingline">
</div>
<div id="lstnumberx49" class="ltx_listingline">
<span id="lstnumberx49.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx49.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span>
</div>
<div id="lstnumberx50" class="ltx_listingline">
<span id="lstnumberx50.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">def</span><span id="lstnumberx50.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx50.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">binary_sum_diff</span><span id="lstnumberx50.4" class="ltx_text ltx_font_typewriter">():</span>
</div>
<div id="lstnumberx51" class="ltx_listingline">
<span id="lstnumberx51.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx51.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx51.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx51.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Convert</span><span id="lstnumberx51.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx51.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">binary</span><span id="lstnumberx51.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx51.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx51.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx51.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">decimal</span>
</div>
<div id="lstnumberx52" class="ltx_listingline">
<span id="lstnumberx52.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx52.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">num1</span><span id="lstnumberx52.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx52.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx52.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx52.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">int</span><span id="lstnumberx52.7" class="ltx_text ltx_font_typewriter">("1011",</span><span id="lstnumberx52.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx52.9" class="ltx_text ltx_font_typewriter">2)</span>
</div>
<div id="lstnumberx53" class="ltx_listingline">
<span id="lstnumberx53.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx53.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">num2</span><span id="lstnumberx53.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx53.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx53.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx53.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">int</span><span id="lstnumberx53.7" class="ltx_text ltx_font_typewriter">("101",</span><span id="lstnumberx53.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx53.9" class="ltx_text ltx_font_typewriter">2)</span>
</div>
<div id="lstnumberx54" class="ltx_listingline">
<span id="lstnumberx54.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx54.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">num3</span><span id="lstnumberx54.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx54.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx54.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx54.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">int</span><span id="lstnumberx54.7" class="ltx_text ltx_font_typewriter">("1100",</span><span id="lstnumberx54.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx54.9" class="ltx_text ltx_font_typewriter">2)</span>
</div>
<div id="lstnumberx55" class="ltx_listingline">
<span id="lstnumberx55.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx55.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">num4</span><span id="lstnumberx55.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx55.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx55.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx55.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">int</span><span id="lstnumberx55.7" class="ltx_text ltx_font_typewriter">("1101",</span><span id="lstnumberx55.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx55.9" class="ltx_text ltx_font_typewriter">2)</span>
</div>
<div id="lstnumberx56" class="ltx_listingline">
</div>
<div id="lstnumberx57" class="ltx_listingline">
<span id="lstnumberx57.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx57.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx57.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx57.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Perform</span><span id="lstnumberx57.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx57.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx57.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx57.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">operations</span>
</div>
<div id="lstnumberx58" class="ltx_listingline">
<span id="lstnumberx58.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx58.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx58.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx58.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx58.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx58.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">num1</span><span id="lstnumberx58.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx58.8" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx58.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx58.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">num2</span><span id="lstnumberx58.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx58.12" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx58.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx58.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">num3</span><span id="lstnumberx58.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx58.16" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx58.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx58.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">num4</span>
</div>
<div id="lstnumberx59" class="ltx_listingline">
</div>
<div id="lstnumberx60" class="ltx_listingline">
<span id="lstnumberx60.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx60.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx60.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx60.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Convert</span><span id="lstnumberx60.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx60.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx60.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx60.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx60.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx60.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">back</span><span id="lstnumberx60.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx60.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx60.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx60.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">binary</span>
</div>
<div id="lstnumberx61" class="ltx_listingline">
<span id="lstnumberx61.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx61.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result_binary</span><span id="lstnumberx61.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx61.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx61.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx61.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">format</span><span id="lstnumberx61.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx61.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx61.9" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx61.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx61.11" class="ltx_text ltx_font_typewriter">"</span><span id="lstnumberx61.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b</span><span id="lstnumberx61.13" class="ltx_text ltx_font_typewriter">")</span>
</div>
<div id="lstnumberx62" class="ltx_listingline">
</div>
<div id="lstnumberx63" class="ltx_listingline">
<span id="lstnumberx63.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx63.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx63.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx63.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result_binary</span>
</div>
<div id="lstnumberx64" class="ltx_listingline">
</div>
<div id="lstnumberx65" class="ltx_listingline">
<span id="lstnumberx65.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx65.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx65.3" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx65.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx65.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">binary_sum_diff</span><span id="lstnumberx65.6" class="ltx_text ltx_font_typewriter">()</span>
</div>
<div id="lstnumberx66" class="ltx_listingline">
<span id="lstnumberx66.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">print</span><span id="lstnumberx66.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx66.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx66.4" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx67" class="ltx_listingline">
<span id="lstnumberx67.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx68" class="ltx_listingline">
<span id="lstnumberx68.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx68.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span>
</div>
<div id="lstnumberx69" class="ltx_listingline">
<span id="lstnumberx69.1" class="ltx_text ltx_font_typewriter">10001</span>
</div>
<div id="lstnumberx70" class="ltx_listingline">
<span id="lstnumberx70.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx71" class="ltx_listingline">
<span id="lstnumberx71.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx71.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx71.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">answer</span><span id="lstnumberx71.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx71.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">expressed</span><span id="lstnumberx71.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx71.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx71.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx71.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">base</span><span id="lstnumberx71.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx71.11" class="ltx_text ltx_font_typewriter">2</span><span id="lstnumberx71.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx71.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx71.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx71.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx71.16" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx71.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx71.18" class="ltx_text ltx_font_typewriter">{10001</span><span id="lstnumberx71.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_2</span><span id="lstnumberx71.20" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx71.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx71.22" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx72" class="ltx_listingline">
</div>
<div id="lstnumberx73" class="ltx_listingline">
<span id="lstnumberx73.1" class="ltx_text ltx_font_typewriter">---</span>
</div>
<div id="lstnumberx74" class="ltx_listingline">
</div>
<div id="lstnumberx75" class="ltx_listingline">
<span id="lstnumberx75.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Question</span><span id="lstnumberx75.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx75.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx75.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Find</span><span id="lstnumberx75.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx75.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx75.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx75.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">matrix</span><span id="lstnumberx75.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx75.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx75.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx75.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">corresponds</span><span id="lstnumberx75.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx75.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx75.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx75.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">projecting</span><span id="lstnumberx75.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx75.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">onto</span><span id="lstnumberx75.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx75.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx75.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx75.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">vector</span><span id="lstnumberx75.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx75.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx75.25" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx75.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">begin</span><span id="lstnumberx75.27" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx75.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pmatrix</span><span id="lstnumberx75.29" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx75.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx75.31" class="ltx_text ltx_font_typewriter">2</span><span id="lstnumberx75.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx75.33" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx75.34" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx75.35" class="ltx_text ltx_font_typewriter">-3</span><span id="lstnumberx75.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx75.37" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx75.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">end</span><span id="lstnumberx75.39" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx75.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pmatrix</span><span id="lstnumberx75.41" class="ltx_text ltx_font_typewriter">}.</span><span id="lstnumberx75.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span>
</div>
<div id="lstnumberx76" class="ltx_listingline">
</div>
<div id="lstnumberx77" class="ltx_listingline">
<span id="lstnumberx77.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solution</span><span id="lstnumberx77.2" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx78" class="ltx_listingline">
<span id="lstnumberx78.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx78.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span>
</div>
<div id="lstnumberx79" class="ltx_listingline">
<span id="lstnumberx79.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx79.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx79.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx79.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx79.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx79.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx79.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Matrix</span><span id="lstnumberx79.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx79.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx79.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">simplify</span>
</div>
<div id="lstnumberx80" class="ltx_listingline">
</div>
<div id="lstnumberx81" class="ltx_listingline">
<span id="lstnumberx81.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">def</span><span id="lstnumberx81.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx81.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">projection_matrix</span><span id="lstnumberx81.4" class="ltx_text ltx_font_typewriter">():</span>
</div>
<div id="lstnumberx82" class="ltx_listingline">
<span id="lstnumberx82.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx82.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">v</span><span id="lstnumberx82.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx82.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx82.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx82.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Matrix</span><span id="lstnumberx82.7" class="ltx_text ltx_font_typewriter">([2,</span><span id="lstnumberx82.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx82.9" class="ltx_text ltx_font_typewriter">-3])</span>
</div>
<div id="lstnumberx83" class="ltx_listingline">
<span id="lstnumberx83.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx83.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">P</span><span id="lstnumberx83.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx83.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx83.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx83.6" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx83.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">v</span><span id="lstnumberx83.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx83.9" class="ltx_text ltx_font_typewriter">*</span><span id="lstnumberx83.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx83.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">v</span><span id="lstnumberx83.12" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx83.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">T</span><span id="lstnumberx83.14" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx83.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx83.16" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx83.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx83.18" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx83.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">v</span><span id="lstnumberx83.20" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx83.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">T</span><span id="lstnumberx83.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx83.23" class="ltx_text ltx_font_typewriter">*</span><span id="lstnumberx83.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx83.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">v</span><span id="lstnumberx83.26" class="ltx_text ltx_font_typewriter">)[0]</span>
</div>
<div id="lstnumberx84" class="ltx_listingline">
</div>
<div id="lstnumberx85" class="ltx_listingline">
<span id="lstnumberx85.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx85.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx85.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx85.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Simplify</span><span id="lstnumberx85.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx85.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx85.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx85.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">elements</span>
</div>
<div id="lstnumberx86" class="ltx_listingline">
<span id="lstnumberx86.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx86.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">P</span><span id="lstnumberx86.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx86.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx86.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx86.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">P</span><span id="lstnumberx86.7" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx86.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">applyfunc</span><span id="lstnumberx86.9" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx86.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">simplify</span><span id="lstnumberx86.11" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx87" class="ltx_listingline">
</div>
<div id="lstnumberx88" class="ltx_listingline">
<span id="lstnumberx88.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx88.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx88.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx88.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">P</span>
</div>
<div id="lstnumberx89" class="ltx_listingline">
</div>
<div id="lstnumberx90" class="ltx_listingline">
<span id="lstnumberx90.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx90.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx90.3" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx90.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx90.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">projection_matrix</span><span id="lstnumberx90.6" class="ltx_text ltx_font_typewriter">()</span>
</div>
<div id="lstnumberx91" class="ltx_listingline">
<span id="lstnumberx91.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">print</span><span id="lstnumberx91.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx91.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx91.4" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx92" class="ltx_listingline">
<span id="lstnumberx92.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx93" class="ltx_listingline">
<span id="lstnumberx93.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx93.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span>
</div>
<div id="lstnumberx94" class="ltx_listingline">
<span id="lstnumberx94.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Matrix</span><span id="lstnumberx94.2" class="ltx_text ltx_font_typewriter">([[4/13,</span><span id="lstnumberx94.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx94.4" class="ltx_text ltx_font_typewriter">-6/13],</span><span id="lstnumberx94.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx94.6" class="ltx_text ltx_font_typewriter">[-6/13,</span><span id="lstnumberx94.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx94.8" class="ltx_text ltx_font_typewriter">9/13]])</span>
</div>
<div id="lstnumberx95" class="ltx_listingline">
<span id="lstnumberx95.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx96" class="ltx_listingline">
</div>
<div id="lstnumberx97" class="ltx_listingline">
<span id="lstnumberx97.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx97.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx97.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">matrix</span><span id="lstnumberx97.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx97.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx97.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx97.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx97.8" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx97.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx97.10" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx97.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">begin</span><span id="lstnumberx97.12" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx97.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pmatrix</span><span id="lstnumberx97.14" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx97.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx97.16" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx97.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx97.18" class="ltx_text ltx_font_typewriter">{4}{13}</span><span id="lstnumberx97.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx97.20" class="ltx_text ltx_font_typewriter">&amp;</span><span id="lstnumberx97.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx97.22" class="ltx_text ltx_font_typewriter">-\</span><span id="lstnumberx97.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx97.24" class="ltx_text ltx_font_typewriter">{6}{13}</span><span id="lstnumberx97.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx97.26" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx97.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx97.28" class="ltx_text ltx_font_typewriter">-\</span><span id="lstnumberx97.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx97.30" class="ltx_text ltx_font_typewriter">{6}{13}</span><span id="lstnumberx97.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx97.32" class="ltx_text ltx_font_typewriter">&amp;</span><span id="lstnumberx97.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx97.34" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx97.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx97.36" class="ltx_text ltx_font_typewriter">{9}{13}</span><span id="lstnumberx97.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx97.38" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx97.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">end</span><span id="lstnumberx97.40" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx97.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pmatrix</span><span id="lstnumberx97.42" class="ltx_text ltx_font_typewriter">}}.</span><span id="lstnumberx97.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span>
</div>
<div id="lstnumberx98" class="ltx_listingline">
</div>
<div id="lstnumberx99" class="ltx_listingline">
<span id="lstnumberx99.1" class="ltx_text ltx_font_typewriter">---</span>
</div>
<div id="lstnumberx100" class="ltx_listingline">
</div>
<div id="lstnumberx101" class="ltx_listingline">
<span id="lstnumberx101.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Question</span><span id="lstnumberx101.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx101.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx101.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solve</span><span id="lstnumberx101.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx101.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx101.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx101.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">inequality</span>
</div>
<div id="lstnumberx102" class="ltx_listingline">
<span id="lstnumberx102.1" class="ltx_text ltx_font_typewriter">\[\</span><span id="lstnumberx102.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx102.3" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx102.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx102.5" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx102.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx102.7" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx102.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx102.9" class="ltx_text ltx_font_typewriter">25}{</span><span id="lstnumberx102.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx102.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx102.12" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx102.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx102.14" class="ltx_text ltx_font_typewriter">5}</span><span id="lstnumberx102.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx102.16" class="ltx_text ltx_font_typewriter">&lt;</span><span id="lstnumberx102.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx102.18" class="ltx_text ltx_font_typewriter">0.\]</span>
</div>
<div id="lstnumberx103" class="ltx_listingline">
</div>
<div id="lstnumberx104" class="ltx_listingline">
<span id="lstnumberx104.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solution</span><span id="lstnumberx104.2" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx105" class="ltx_listingline">
<span id="lstnumberx105.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Let</span><span id="lstnumberx105.2" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx105.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx105.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx105.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">use</span><span id="lstnumberx105.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx105.7" class="ltx_text ltx_font_typewriter">‘</span><span id="lstnumberx105.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx105.9" class="ltx_text ltx_font_typewriter">‘</span><span id="lstnumberx105.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx105.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx105.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx105.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve</span><span id="lstnumberx105.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx105.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx105.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx105.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">inequality</span><span id="lstnumberx105.18" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx106" class="ltx_listingline">
</div>
<div id="lstnumberx107" class="ltx_listingline">
<span id="lstnumberx107.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx107.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span>
</div>
<div id="lstnumberx108" class="ltx_listingline">
<span id="lstnumberx108.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx108.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx108.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx108.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx108.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx108.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx108.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span><span id="lstnumberx108.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx108.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx108.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">simplify</span>
</div>
<div id="lstnumberx109" class="ltx_listingline">
<span id="lstnumberx109.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx109.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx109.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx109.4" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx109.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solvers</span><span id="lstnumberx109.6" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx109.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">inequalities</span><span id="lstnumberx109.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx109.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx109.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx109.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve_univariate_inequality</span>
</div>
<div id="lstnumberx110" class="ltx_listingline">
<span id="lstnumberx110.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx110.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx110.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx110.4" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx110.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">core</span><span id="lstnumberx110.6" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx110.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">relational</span><span id="lstnumberx110.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx110.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx110.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx110.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">StrictLessThan</span>
</div>
<div id="lstnumberx111" class="ltx_listingline">
</div>
<div id="lstnumberx112" class="ltx_listingline">
<span id="lstnumberx112.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">def</span><span id="lstnumberx112.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx112.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve_inequality</span><span id="lstnumberx112.4" class="ltx_text ltx_font_typewriter">():</span>
</div>
<div id="lstnumberx113" class="ltx_listingline">
<span id="lstnumberx113.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx113.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx113.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx113.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx113.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx113.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span><span id="lstnumberx113.7" class="ltx_text ltx_font_typewriter">(’</span><span id="lstnumberx113.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx113.9" class="ltx_text ltx_font_typewriter">’)</span>
</div>
<div id="lstnumberx114" class="ltx_listingline">
<span id="lstnumberx114.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx114.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">expression</span><span id="lstnumberx114.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx114.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx114.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx114.6" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx114.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx114.8" class="ltx_text ltx_font_typewriter">**2</span><span id="lstnumberx114.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx114.10" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx114.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx114.12" class="ltx_text ltx_font_typewriter">25)</span><span id="lstnumberx114.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx114.14" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx114.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx114.16" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx114.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx114.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx114.19" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx114.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx114.21" class="ltx_text ltx_font_typewriter">5)</span>
</div>
<div id="lstnumberx115" class="ltx_listingline">
<span id="lstnumberx115.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx115.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">inequality</span><span id="lstnumberx115.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx115.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx115.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx115.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">StrictLessThan</span><span id="lstnumberx115.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx115.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">expression</span><span id="lstnumberx115.9" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx115.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx115.11" class="ltx_text ltx_font_typewriter">0)</span>
</div>
<div id="lstnumberx116" class="ltx_listingline">
<span id="lstnumberx116.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx116.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solution</span><span id="lstnumberx116.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx116.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx116.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx116.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve_univariate_inequality</span><span id="lstnumberx116.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx116.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">inequality</span><span id="lstnumberx116.9" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx116.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx116.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx116.12" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx116.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx116.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">relational</span><span id="lstnumberx116.15" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx116.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">False</span><span id="lstnumberx116.17" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx117" class="ltx_listingline">
<span id="lstnumberx117.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx117.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">simplified_solution</span><span id="lstnumberx117.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx117.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx117.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx117.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">simplify</span><span id="lstnumberx117.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx117.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solution</span><span id="lstnumberx117.9" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx118" class="ltx_listingline">
</div>
<div id="lstnumberx119" class="ltx_listingline">
<span id="lstnumberx119.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx119.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx119.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx119.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">simplified_solution</span>
</div>
<div id="lstnumberx120" class="ltx_listingline">
</div>
<div id="lstnumberx121" class="ltx_listingline">
<span id="lstnumberx121.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx121.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx121.3" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx121.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx121.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve_inequality</span><span id="lstnumberx121.6" class="ltx_text ltx_font_typewriter">()</span>
</div>
<div id="lstnumberx122" class="ltx_listingline">
<span id="lstnumberx122.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">print</span><span id="lstnumberx122.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx122.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx122.4" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx123" class="ltx_listingline">
<span id="lstnumberx123.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx124" class="ltx_listingline">
<span id="lstnumberx124.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx124.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span>
</div>
<div id="lstnumberx125" class="ltx_listingline">
<span id="lstnumberx125.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Union</span><span id="lstnumberx125.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx125.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Interval</span><span id="lstnumberx125.4" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx125.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">open</span><span id="lstnumberx125.6" class="ltx_text ltx_font_typewriter">(-</span><span id="lstnumberx125.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">oo</span><span id="lstnumberx125.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx125.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx125.10" class="ltx_text ltx_font_typewriter">-5),</span><span id="lstnumberx125.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx125.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Interval</span><span id="lstnumberx125.13" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx125.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">open</span><span id="lstnumberx125.15" class="ltx_text ltx_font_typewriter">(-5,</span><span id="lstnumberx125.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx125.17" class="ltx_text ltx_font_typewriter">5))</span>
</div>
<div id="lstnumberx126" class="ltx_listingline">
<span id="lstnumberx126.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx127" class="ltx_listingline">
<span id="lstnumberx127.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx127.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx127.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">answer</span><span id="lstnumberx127.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx127.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx127.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx127.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx127.8" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx127.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx127.10" class="ltx_text ltx_font_typewriter">{(-\</span><span id="lstnumberx127.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">infty</span><span id="lstnumberx127.12" class="ltx_text ltx_font_typewriter">,-5)\</span><span id="lstnumberx127.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cup</span><span id="lstnumberx127.14" class="ltx_text ltx_font_typewriter">(-5,5)}</span><span id="lstnumberx127.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span>
</div>
<div id="lstnumberx128" class="ltx_listingline">
</div>
<div id="lstnumberx129" class="ltx_listingline">
<span id="lstnumberx129.1" class="ltx_text ltx_font_typewriter">---</span>
</div>
</div>
</figure>
</section>
<section id="A6" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix F </span>Examples</h2>

<section id="A6.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">F.1 </span>Success Cases</h3>

<figure id="LST2" class="ltx_float ltx_lstlisting">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float">Listing 2: </span>Success case for <span id="LST2.3.1" class="ltx_text ltx_font_smallcaps">ToRA</span>: <span id="LST2.4.2" class="ltx_text ltx_font_bold">Rationale for planning</span>.</figcaption>
<div id="LST2.5" class="ltx_listing ltx_lstlisting ltx_framed ltx_framed_topbottom ltx_listing">
<div class="ltx_listing_data"><a href="data:text/plain;base64,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" download="section/cases/success/plan.md">⬇</a></div>
<div id="lstnumberx130" class="ltx_listingline">
<span id="lstnumberx130.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Question</span><span id="lstnumberx130.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx130.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx130.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">What</span><span id="lstnumberx130.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx130.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx130.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx130.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">values</span><span id="lstnumberx130.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx130.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx130.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx130.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x$</span><span id="lstnumberx130.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx130.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx130.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx130.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">not</span><span id="lstnumberx130.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx130.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx130.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx130.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx130.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx130.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">domain</span><span id="lstnumberx130.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx130.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx130.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx130.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$f</span><span id="lstnumberx130.27" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx130.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx130.29" class="ltx_text ltx_font_typewriter">)=\</span><span id="lstnumberx130.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx130.31" class="ltx_text ltx_font_typewriter">{1}{|</span><span id="lstnumberx130.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx130.33" class="ltx_text ltx_font_typewriter">^2-6</span><span id="lstnumberx130.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx130.35" class="ltx_text ltx_font_typewriter">+8|+|</span><span id="lstnumberx130.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx130.37" class="ltx_text ltx_font_typewriter">^2+</span><span id="lstnumberx130.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx130.39" class="ltx_text ltx_font_typewriter">-6|}</span><span id="lstnumberx130.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx130.41" class="ltx_text ltx_font_typewriter">?</span>
</div>
<div id="lstnumberx131" class="ltx_listingline">
</div>
<div id="lstnumberx132" class="ltx_listingline">
<span id="lstnumberx132.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solution</span><span id="lstnumberx132.2" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx133" class="ltx_listingline">
<span id="lstnumberx133.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx133.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">domain</span><span id="lstnumberx133.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx133.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx133.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">function</span><span id="lstnumberx133.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx133.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">all</span><span id="lstnumberx133.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx133.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">numbers</span><span id="lstnumberx133.18" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx133.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">so</span><span id="lstnumberx133.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx133.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">need</span><span id="lstnumberx133.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx133.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">find</span><span id="lstnumberx133.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx133.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">values</span><span id="lstnumberx133.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx133.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx133.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx133.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">which</span><span id="lstnumberx133.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx133.43" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">denominator</span><span id="lstnumberx133.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx133.47" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.48" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx133.49" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.50" class="ltx_text ltx_lst_identifier ltx_font_typewriter">function</span><span id="lstnumberx133.51" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx133.53" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx133.54" class="ltx_text ltx_lst_identifier ltx_font_typewriter">zero</span><span id="lstnumberx133.55" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx134" class="ltx_listingline">
<span id="lstnumberx134.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx134.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">denominator</span><span id="lstnumberx134.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx134.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx134.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">function</span><span id="lstnumberx134.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx134.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx134.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx134.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx134.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx134.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">absolute</span><span id="lstnumberx134.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">values</span><span id="lstnumberx134.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx134.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx134.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">two</span><span id="lstnumberx134.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">terms</span><span id="lstnumberx134.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">inside</span><span id="lstnumberx134.34" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx134.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">absolute</span><span id="lstnumberx134.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx134.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">value</span><span id="lstnumberx134.40" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx135" class="ltx_listingline">
<span id="lstnumberx135.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Each</span><span id="lstnumberx135.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx135.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">these</span><span id="lstnumberx135.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">terms</span><span id="lstnumberx135.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx135.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx135.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">quadratic</span><span id="lstnumberx135.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx135.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx135.18" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx135.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx135.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx135.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots</span><span id="lstnumberx135.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx135.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx135.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">quadratic</span><span id="lstnumberx135.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equation</span><span id="lstnumberx135.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx135.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx135.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">unless</span><span id="lstnumberx135.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx135.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equation</span><span id="lstnumberx135.43" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx135.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">identically</span><span id="lstnumberx135.47" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx135.48" class="ltx_text ltx_lst_identifier ltx_font_typewriter">zero</span><span id="lstnumberx135.49" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx136" class="ltx_listingline">
<span id="lstnumberx136.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">So</span><span id="lstnumberx136.2" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx136.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx136.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx136.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx136.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">need</span><span id="lstnumberx136.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx136.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx136.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx136.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">find</span><span id="lstnumberx136.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx136.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx136.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx136.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">values</span><span id="lstnumberx136.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx136.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx136.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx136.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx136.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx136.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx136.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx136.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">which</span><span id="lstnumberx136.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx136.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx136.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx136.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">two</span><span id="lstnumberx136.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx136.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">quadratic</span><span id="lstnumberx136.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx136.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equations</span><span id="lstnumberx136.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx136.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx136.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx136.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">identically</span><span id="lstnumberx136.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx136.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">zero</span><span id="lstnumberx136.37" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx137" class="ltx_listingline">
</div>
<div id="lstnumberx138" class="ltx_listingline">
<span id="lstnumberx138.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx138.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span>
</div>
<div id="lstnumberx139" class="ltx_listingline">
<span id="lstnumberx139.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx139.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx139.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx139.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx139.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx139.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx139.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span><span id="lstnumberx139.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx139.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx139.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Abs</span><span id="lstnumberx139.11" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx139.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx139.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solveset</span><span id="lstnumberx139.14" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx139.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx139.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">S</span>
</div>
<div id="lstnumberx140" class="ltx_listingline">
<span id="lstnumberx140.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx140.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx140.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx140.4" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx140.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">core</span><span id="lstnumberx140.6" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx140.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">relational</span><span id="lstnumberx140.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx140.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx140.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx140.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Ne</span>
</div>
<div id="lstnumberx141" class="ltx_listingline">
</div>
<div id="lstnumberx142" class="ltx_listingline">
<span id="lstnumberx142.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">def</span><span id="lstnumberx142.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx142.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">non_domain_values</span><span id="lstnumberx142.4" class="ltx_text ltx_font_typewriter">():</span>
</div>
<div id="lstnumberx143" class="ltx_listingline">
<span id="lstnumberx143.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx143.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx143.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx143.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx143.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx143.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span><span id="lstnumberx143.7" class="ltx_text ltx_font_typewriter">(’</span><span id="lstnumberx143.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx143.9" class="ltx_text ltx_font_typewriter">’)</span>
</div>
<div id="lstnumberx144" class="ltx_listingline">
<span id="lstnumberx144.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx144.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx144.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx144.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">These</span><span id="lstnumberx144.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx144.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx144.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx144.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx144.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx144.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">values</span><span id="lstnumberx144.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx144.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx144.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx144.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx144.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx144.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx144.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx144.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">which</span><span id="lstnumberx144.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx144.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx144.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx144.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">function</span><span id="lstnumberx144.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx144.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx144.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx144.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">not</span><span id="lstnumberx144.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx144.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">defined</span>
</div>
<div id="lstnumberx145" class="ltx_listingline">
<span id="lstnumberx145.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx145.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots1</span><span id="lstnumberx145.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx145.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx145.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx145.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solveset</span><span id="lstnumberx145.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx145.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Ne</span><span id="lstnumberx145.9" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx145.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx145.11" class="ltx_text ltx_font_typewriter">**2</span><span id="lstnumberx145.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx145.13" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx145.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx145.15" class="ltx_text ltx_font_typewriter">6*</span><span id="lstnumberx145.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx145.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx145.18" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx145.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx145.20" class="ltx_text ltx_font_typewriter">8,</span><span id="lstnumberx145.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx145.22" class="ltx_text ltx_font_typewriter">0),</span><span id="lstnumberx145.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx145.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx145.25" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx145.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx145.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">domain</span><span id="lstnumberx145.28" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx145.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">S</span><span id="lstnumberx145.30" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx145.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Reals</span><span id="lstnumberx145.32" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx146" class="ltx_listingline">
<span id="lstnumberx146.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx146.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots2</span><span id="lstnumberx146.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx146.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx146.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx146.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solveset</span><span id="lstnumberx146.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx146.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Ne</span><span id="lstnumberx146.9" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx146.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx146.11" class="ltx_text ltx_font_typewriter">**2</span><span id="lstnumberx146.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx146.13" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx146.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx146.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx146.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx146.17" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx146.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx146.19" class="ltx_text ltx_font_typewriter">6,</span><span id="lstnumberx146.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx146.21" class="ltx_text ltx_font_typewriter">0),</span><span id="lstnumberx146.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx146.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx146.24" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx146.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx146.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">domain</span><span id="lstnumberx146.27" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx146.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">S</span><span id="lstnumberx146.29" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx146.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Reals</span><span id="lstnumberx146.31" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx147" class="ltx_listingline">
<span id="lstnumberx147.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx147.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx147.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx147.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots</span><span id="lstnumberx147.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx147.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx147.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">quadratic</span><span id="lstnumberx147.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equation</span><span id="lstnumberx147.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx147.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">always</span><span id="lstnumberx147.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx147.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx147.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">domain</span><span id="lstnumberx147.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx147.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx147.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">function</span><span id="lstnumberx147.31" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx147.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">so</span><span id="lstnumberx147.34" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx147.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">need</span><span id="lstnumberx147.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx147.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">find</span><span id="lstnumberx147.42" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx147.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">values</span><span id="lstnumberx147.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx147.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx147.50" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx147.52" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.53" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx147.54" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">not</span><span id="lstnumberx147.56" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx147.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx147.60" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">union</span><span id="lstnumberx147.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx147.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.65" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx147.66" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx147.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots</span>
</div>
<div id="lstnumberx148" class="ltx_listingline">
<span id="lstnumberx148.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx148.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">non_domain_values</span><span id="lstnumberx148.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx148.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx148.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx148.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">S</span><span id="lstnumberx148.7" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx148.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Reals</span><span id="lstnumberx148.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx148.10" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx148.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx148.12" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx148.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots1</span><span id="lstnumberx148.14" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx148.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">union</span><span id="lstnumberx148.16" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx148.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots2</span><span id="lstnumberx148.18" class="ltx_text ltx_font_typewriter">))</span>
</div>
<div id="lstnumberx149" class="ltx_listingline">
</div>
<div id="lstnumberx150" class="ltx_listingline">
<span id="lstnumberx150.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx150.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx150.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx150.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">non_domain_values</span>
</div>
<div id="lstnumberx151" class="ltx_listingline">
</div>
<div id="lstnumberx152" class="ltx_listingline">
<span id="lstnumberx152.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx152.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx152.3" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx152.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx152.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">non_domain_values</span><span id="lstnumberx152.6" class="ltx_text ltx_font_typewriter">()</span>
</div>
<div id="lstnumberx153" class="ltx_listingline">
<span id="lstnumberx153.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">print</span><span id="lstnumberx153.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx153.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx153.4" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx154" class="ltx_listingline">
<span id="lstnumberx154.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx155" class="ltx_listingline">
<span id="lstnumberx155.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx155.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span>
</div>
<div id="lstnumberx156" class="ltx_listingline">
<span id="lstnumberx156.1" class="ltx_text ltx_font_typewriter">{2}</span>
</div>
<div id="lstnumberx157" class="ltx_listingline">
<span id="lstnumberx157.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx158" class="ltx_listingline">
<span id="lstnumberx158.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx158.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx158.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx158.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx158.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">value</span><span id="lstnumberx158.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx158.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx158.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx158.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x$</span><span id="lstnumberx158.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx158.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx158.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx158.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx158.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx158.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">not</span><span id="lstnumberx158.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx158.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx158.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx158.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx158.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx158.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">domain</span><span id="lstnumberx158.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx158.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx158.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx158.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx158.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx158.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">function</span><span id="lstnumberx158.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx158.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx158.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx158.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx158.32" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx158.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx158.34" class="ltx_text ltx_font_typewriter">{2}</span><span id="lstnumberx158.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx158.36" class="ltx_text ltx_font_typewriter">.</span>
</div>
</div>
</figure>
<figure id="LST3" class="ltx_float ltx_lstlisting">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float">Listing 3: </span>Success case for <span id="LST3.3.1" class="ltx_text ltx_font_smallcaps">ToRA</span>: <span id="LST3.4.2" class="ltx_text ltx_font_bold">Self-Correcting Errors with tool feedback</span>.</figcaption>
<div id="LST3.5" class="ltx_listing ltx_lstlisting ltx_framed ltx_framed_topbottom ltx_listing">
<div class="ltx_listing_data"><a href="data:text/plain;base64,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" download="section/cases/success/correct.md">⬇</a></div>
<div id="lstnumberx159" class="ltx_listingline">
<span id="lstnumberx159.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Question</span><span id="lstnumberx159.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx159.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx159.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">difference</span><span id="lstnumberx159.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">between</span><span id="lstnumberx159.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">two</span><span id="lstnumberx159.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">perfect</span><span id="lstnumberx159.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">squares</span><span id="lstnumberx159.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx159.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.18" class="ltx_text ltx_font_typewriter">133.</span><span id="lstnumberx159.19" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx159.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">What</span><span id="lstnumberx159.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx159.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx159.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">smallest</span><span id="lstnumberx159.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">possible</span><span id="lstnumberx159.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx159.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx159.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx159.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">two</span><span id="lstnumberx159.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">perfect</span><span id="lstnumberx159.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx159.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">squares</span><span id="lstnumberx159.41" class="ltx_text ltx_font_typewriter">?</span>
</div>
<div id="lstnumberx160" class="ltx_listingline">
</div>
<div id="lstnumberx161" class="ltx_listingline">
<span id="lstnumberx161.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solution</span><span id="lstnumberx161.2" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx162" class="ltx_listingline">
<span id="lstnumberx162.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">We</span><span id="lstnumberx162.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">can</span><span id="lstnumberx162.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve</span><span id="lstnumberx162.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">this</span><span id="lstnumberx162.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">problem</span><span id="lstnumberx162.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">by</span><span id="lstnumberx162.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">noting</span><span id="lstnumberx162.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx162.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">any</span><span id="lstnumberx162.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">perfect</span><span id="lstnumberx162.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">square</span><span id="lstnumberx162.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">can</span><span id="lstnumberx162.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">be</span><span id="lstnumberx162.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">written</span><span id="lstnumberx162.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">as</span><span id="lstnumberx162.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx162.32" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx162.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx162.34" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.35" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx162.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b</span><span id="lstnumberx162.38" class="ltx_text ltx_font_typewriter">)^2</span><span id="lstnumberx162.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx162.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">or</span><span id="lstnumberx162.42" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx162.44" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx162.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx162.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.47" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx162.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b</span><span id="lstnumberx162.50" class="ltx_text ltx_font_typewriter">)^2</span><span id="lstnumberx162.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx162.52" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.53" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx162.54" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">some</span><span id="lstnumberx162.56" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">integers</span><span id="lstnumberx162.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$a$</span><span id="lstnumberx162.60" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx162.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$b$</span><span id="lstnumberx162.64" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx162.65" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.66" class="ltx_text ltx_lst_identifier ltx_font_typewriter">So</span><span id="lstnumberx162.67" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.68" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx162.69" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.70" class="ltx_text ltx_lst_identifier ltx_font_typewriter">can</span><span id="lstnumberx162.71" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.72" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rewrite</span><span id="lstnumberx162.73" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.74" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx162.75" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.76" class="ltx_text ltx_lst_identifier ltx_font_typewriter">question</span><span id="lstnumberx162.77" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.78" class="ltx_text ltx_lst_identifier ltx_font_typewriter">as</span><span id="lstnumberx162.79" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.80" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx162.81" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.82" class="ltx_text ltx_lst_identifier ltx_font_typewriter">following</span><span id="lstnumberx162.83" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.84" class="ltx_text ltx_lst_identifier ltx_font_typewriter">problem</span><span id="lstnumberx162.85" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx162.86" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.87" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Find</span><span id="lstnumberx162.88" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.89" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx162.90" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.91" class="ltx_text ltx_lst_identifier ltx_font_typewriter">minimum</span><span id="lstnumberx162.92" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.93" class="ltx_text ltx_lst_identifier ltx_font_typewriter">possible</span><span id="lstnumberx162.94" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.95" class="ltx_text ltx_lst_identifier ltx_font_typewriter">value</span><span id="lstnumberx162.96" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.97" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx162.98" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.99" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx162.100" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx162.101" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx162.102" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.103" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx162.104" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.105" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b</span><span id="lstnumberx162.106" class="ltx_text ltx_font_typewriter">)^2</span><span id="lstnumberx162.107" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.108" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx162.109" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.110" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx162.111" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx162.112" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.113" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx162.114" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.115" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b</span><span id="lstnumberx162.116" class="ltx_text ltx_font_typewriter">)^2</span><span id="lstnumberx162.117" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx162.118" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.119" class="ltx_text ltx_lst_identifier ltx_font_typewriter">given</span><span id="lstnumberx162.120" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.121" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx162.122" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.123" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx162.124" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx162.125" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx162.126" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.127" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx162.128" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.129" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b</span><span id="lstnumberx162.130" class="ltx_text ltx_font_typewriter">)^2</span><span id="lstnumberx162.131" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.132" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx162.133" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.134" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx162.135" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx162.136" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.137" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx162.138" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.139" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b</span><span id="lstnumberx162.140" class="ltx_text ltx_font_typewriter">)^2</span><span id="lstnumberx162.141" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.142" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx162.143" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx162.144" class="ltx_text ltx_font_typewriter">133</span><span id="lstnumberx162.145" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx162.146" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx163" class="ltx_listingline">
</div>
<div id="lstnumberx164" class="ltx_listingline">
<span id="lstnumberx164.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx164.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span>
</div>
<div id="lstnumberx165" class="ltx_listingline">
<span id="lstnumberx165.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx165.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx165.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx165.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx165.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx165.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx165.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span><span id="lstnumberx165.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx165.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx165.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Eq</span><span id="lstnumberx165.11" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx165.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx165.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve</span>
</div>
<div id="lstnumberx166" class="ltx_listingline">
</div>
<div id="lstnumberx167" class="ltx_listingline">
<span id="lstnumberx167.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">def</span><span id="lstnumberx167.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx167.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">min_sum_perfect_squares</span><span id="lstnumberx167.4" class="ltx_text ltx_font_typewriter">():</span>
</div>
<div id="lstnumberx168" class="ltx_listingline">
<span id="lstnumberx168.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx168.2" class="ltx_text ltx_font_typewriter">"""</span>
</div>
<div id="lstnumberx169" class="ltx_listingline">
<span id="lstnumberx169.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx169.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Find</span><span id="lstnumberx169.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx169.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx169.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx169.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">smallest</span><span id="lstnumberx169.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx169.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">possible</span><span id="lstnumberx169.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx169.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx169.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx169.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx169.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx169.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">two</span><span id="lstnumberx169.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx169.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">perfect</span><span id="lstnumberx169.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx169.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">squares</span>
</div>
<div id="lstnumberx170" class="ltx_listingline">
<span id="lstnumberx170.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx170.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">given</span><span id="lstnumberx170.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx170.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx170.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx170.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx170.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx170.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">difference</span><span id="lstnumberx170.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx170.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">between</span><span id="lstnumberx170.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx170.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">them</span><span id="lstnumberx170.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx170.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx170.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx170.16" class="ltx_text ltx_font_typewriter">133.</span>
</div>
<div id="lstnumberx171" class="ltx_listingline">
</div>
<div id="lstnumberx172" class="ltx_listingline">
<span id="lstnumberx172.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx172.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Returns</span><span id="lstnumberx172.3" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx173" class="ltx_listingline">
<span id="lstnumberx173.1" class="ltx_text ltx_lst_space ltx_font_typewriter">        </span><span id="lstnumberx173.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Integer</span><span id="lstnumberx173.3" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx173.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx173.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx173.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx173.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">minimum</span><span id="lstnumberx173.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx173.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">possible</span><span id="lstnumberx173.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx173.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx173.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx173.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx173.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx173.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx173.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx173.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">two</span><span id="lstnumberx173.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx173.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">perfect</span><span id="lstnumberx173.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx173.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">squares</span><span id="lstnumberx173.22" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx174" class="ltx_listingline">
<span id="lstnumberx174.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx174.2" class="ltx_text ltx_font_typewriter">"""</span>
</div>
<div id="lstnumberx175" class="ltx_listingline">
<span id="lstnumberx175.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx175.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx175.3" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx175.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx175.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b</span><span id="lstnumberx175.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx175.7" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx175.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx175.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span><span id="lstnumberx175.10" class="ltx_text ltx_font_typewriter">(’</span><span id="lstnumberx175.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx175.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx175.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b</span><span id="lstnumberx175.14" class="ltx_text ltx_font_typewriter">’)</span>
</div>
<div id="lstnumberx176" class="ltx_listingline">
</div>
<div id="lstnumberx177" class="ltx_listingline">
<span id="lstnumberx177.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx177.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx177.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx177.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Equation</span><span id="lstnumberx177.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx177.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx177.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx177.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx177.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx177.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">difference</span><span id="lstnumberx177.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx177.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx177.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx177.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx177.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx177.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">squares</span>
</div>
<div id="lstnumberx178" class="ltx_listingline">
<span id="lstnumberx178.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx178.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">diff_eq</span><span id="lstnumberx178.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx178.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx178.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx178.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Eq</span><span id="lstnumberx178.7" class="ltx_text ltx_font_typewriter">((</span><span id="lstnumberx178.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx178.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx178.10" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx178.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx178.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b</span><span id="lstnumberx178.13" class="ltx_text ltx_font_typewriter">)**2</span><span id="lstnumberx178.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx178.15" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx178.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx178.17" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx178.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx178.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx178.20" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx178.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx178.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b</span><span id="lstnumberx178.23" class="ltx_text ltx_font_typewriter">)**2,</span><span id="lstnumberx178.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx178.25" class="ltx_text ltx_font_typewriter">133)</span>
</div>
<div id="lstnumberx179" class="ltx_listingline">
</div>
<div id="lstnumberx180" class="ltx_listingline">
<span id="lstnumberx180.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx180.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx180.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx180.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solve</span><span id="lstnumberx180.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx180.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx180.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx180.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equation</span><span id="lstnumberx180.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx180.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx180.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx180.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b</span>
</div>
<div id="lstnumberx181" class="ltx_listingline">
<span id="lstnumberx181.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx181.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b_solutions</span><span id="lstnumberx181.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx181.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx181.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx181.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve</span><span id="lstnumberx181.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx181.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">diff_eq</span><span id="lstnumberx181.9" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx181.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx181.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b</span><span id="lstnumberx181.12" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx182" class="ltx_listingline">
</div>
<div id="lstnumberx183" class="ltx_listingline">
<span id="lstnumberx183.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx183.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx183.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx183.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Calculate</span><span id="lstnumberx183.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx183.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx183.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx183.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx183.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx183.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx183.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx183.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx183.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx183.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">squares</span><span id="lstnumberx183.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx183.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx183.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx183.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">each</span><span id="lstnumberx183.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx183.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solution</span><span id="lstnumberx183.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx183.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx183.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx183.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b</span>
</div>
<div id="lstnumberx184" class="ltx_listingline">
<span id="lstnumberx184.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx184.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_squares</span><span id="lstnumberx184.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx184.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx184.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx184.6" class="ltx_text ltx_font_typewriter">[(</span><span id="lstnumberx184.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx184.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx184.9" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx184.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx184.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b_sol</span><span id="lstnumberx184.12" class="ltx_text ltx_font_typewriter">)**2</span><span id="lstnumberx184.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx184.14" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx184.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx184.16" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx184.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx184.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx184.19" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx184.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx184.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b_sol</span><span id="lstnumberx184.22" class="ltx_text ltx_font_typewriter">)**2</span><span id="lstnumberx184.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx184.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx184.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx184.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b_sol</span><span id="lstnumberx184.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx184.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx184.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx184.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b_solutions</span><span id="lstnumberx184.31" class="ltx_text ltx_font_typewriter">]</span>
</div>
<div id="lstnumberx185" class="ltx_listingline">
</div>
<div id="lstnumberx186" class="ltx_listingline">
<span id="lstnumberx186.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx186.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx186.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx186.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Find</span><span id="lstnumberx186.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx186.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx186.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx186.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">minimum</span><span id="lstnumberx186.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx186.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">possible</span><span id="lstnumberx186.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx186.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx186.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx186.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx186.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx186.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx186.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx186.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">squares</span>
</div>
<div id="lstnumberx187" class="ltx_listingline">
<span id="lstnumberx187.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx187.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">min_sum_squares</span><span id="lstnumberx187.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx187.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx187.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx187.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">min</span><span id="lstnumberx187.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx187.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_squares</span><span id="lstnumberx187.9" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx188" class="ltx_listingline">
</div>
<div id="lstnumberx189" class="ltx_listingline">
<span id="lstnumberx189.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx189.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx189.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx189.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">min_sum_squares</span>
</div>
<div id="lstnumberx190" class="ltx_listingline">
</div>
<div id="lstnumberx191" class="ltx_listingline">
<span id="lstnumberx191.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx191.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx191.3" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx191.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx191.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">min_sum_perfect_squares</span><span id="lstnumberx191.6" class="ltx_text ltx_font_typewriter">()</span>
</div>
<div id="lstnumberx192" class="ltx_listingline">
<span id="lstnumberx192.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">print</span><span id="lstnumberx192.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx192.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx192.4" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx193" class="ltx_listingline">
<span id="lstnumberx193.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx194" class="ltx_listingline">
<span id="lstnumberx194.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx194.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span>
</div>
<div id="lstnumberx195" class="ltx_listingline">
<span id="lstnumberx195.1" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx195.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx195.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx195.4" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx195.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx195.6" class="ltx_text ltx_font_typewriter">133/(4*</span><span id="lstnumberx195.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx195.8" class="ltx_text ltx_font_typewriter">))**2</span><span id="lstnumberx195.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx195.10" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx195.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx195.12" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx195.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx195.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx195.15" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx195.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx195.17" class="ltx_text ltx_font_typewriter">133/(4*</span><span id="lstnumberx195.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx195.19" class="ltx_text ltx_font_typewriter">))**2</span>
</div>
<div id="lstnumberx196" class="ltx_listingline">
<span id="lstnumberx196.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx197" class="ltx_listingline">
</div>
<div id="lstnumberx198" class="ltx_listingline">
<span id="lstnumberx198.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx198.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx198.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">problem</span><span id="lstnumberx198.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx198.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">seems</span><span id="lstnumberx198.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx198.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx198.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx198.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">be</span><span id="lstnumberx198.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx198.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">more</span><span id="lstnumberx198.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx198.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">complex</span><span id="lstnumberx198.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx198.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">than</span><span id="lstnumberx198.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx198.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">anticipated</span><span id="lstnumberx198.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx198.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx198.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx198.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx198.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx198.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solution</span><span id="lstnumberx198.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx198.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">was</span><span id="lstnumberx198.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx198.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">incorrect</span><span id="lstnumberx198.28" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx199" class="ltx_listingline">
<span id="lstnumberx199.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Since</span><span id="lstnumberx199.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx199.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx199.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx199.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">difference</span><span id="lstnumberx199.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx199.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">between</span><span id="lstnumberx199.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx199.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">two</span><span id="lstnumberx199.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx199.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">squares</span><span id="lstnumberx199.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx199.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">can</span><span id="lstnumberx199.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx199.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">be</span><span id="lstnumberx199.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx199.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">factored</span><span id="lstnumberx199.18" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx199.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx199.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx199.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx199.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">can</span><span id="lstnumberx199.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx199.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">write</span><span id="lstnumberx199.25" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx200" class="ltx_listingline">
</div>
<div id="lstnumberx201" class="ltx_listingline">
<span id="lstnumberx201.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x</span><span id="lstnumberx201.2" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx201.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx201.4" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx201.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx201.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx201.7" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx201.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx201.9" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx201.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx201.11" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx201.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx201.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx201.14" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx201.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx201.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx201.17" class="ltx_text ltx_font_typewriter">)(</span><span id="lstnumberx201.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx201.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx201.20" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx201.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx201.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx201.23" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx201.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span>
</div>
<div id="lstnumberx202" class="ltx_listingline">
</div>
<div id="lstnumberx203" class="ltx_listingline">
<span id="lstnumberx203.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$133$</span><span id="lstnumberx203.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx203.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">has</span><span id="lstnumberx203.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx203.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">two</span><span id="lstnumberx203.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx203.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pairs</span><span id="lstnumberx203.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx203.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx203.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx203.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">factors</span><span id="lstnumberx203.12" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx203.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx203.14" class="ltx_text ltx_font_typewriter">1</span><span id="lstnumberx203.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx203.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx203.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx203.18" class="ltx_text ltx_font_typewriter">133,</span><span id="lstnumberx203.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx203.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx203.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx203.22" class="ltx_text ltx_font_typewriter">7</span><span id="lstnumberx203.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx203.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx203.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx203.26" class="ltx_text ltx_font_typewriter">19.</span>
</div>
<div id="lstnumberx204" class="ltx_listingline">
</div>
<div id="lstnumberx205" class="ltx_listingline">
<span id="lstnumberx205.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">So</span><span id="lstnumberx205.2" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx205.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx205.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">either</span><span id="lstnumberx205.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx205.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x</span><span id="lstnumberx205.7" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx205.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx205.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx205.10" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx205.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx205.12" class="ltx_text ltx_font_typewriter">133</span><span id="lstnumberx205.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx205.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx205.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx205.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx205.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x</span><span id="lstnumberx205.18" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx205.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx205.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx205.21" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx205.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx205.23" class="ltx_text ltx_font_typewriter">1</span><span id="lstnumberx205.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx205.25" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx205.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx205.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">or</span><span id="lstnumberx205.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx205.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x</span><span id="lstnumberx205.30" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx205.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx205.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx205.33" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx205.34" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx205.35" class="ltx_text ltx_font_typewriter">19</span><span id="lstnumberx205.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx205.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx205.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx205.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx205.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x</span><span id="lstnumberx205.41" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx205.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx205.43" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx205.44" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx205.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx205.46" class="ltx_text ltx_font_typewriter">7</span><span id="lstnumberx205.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx205.48" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx206" class="ltx_listingline">
</div>
<div id="lstnumberx207" class="ltx_listingline">
<span id="lstnumberx207.1" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx207.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">For</span><span id="lstnumberx207.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.5" class="ltx_text ltx_font_typewriter">(133,</span><span id="lstnumberx207.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.7" class="ltx_text ltx_font_typewriter">1),</span><span id="lstnumberx207.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x</span><span id="lstnumberx207.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.11" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx207.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.13" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx207.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx207.15" class="ltx_text ltx_font_typewriter">{133</span><span id="lstnumberx207.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.17" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx207.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.19" class="ltx_text ltx_font_typewriter">1}{2}</span><span id="lstnumberx207.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.21" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx207.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.23" class="ltx_text ltx_font_typewriter">67</span><span id="lstnumberx207.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx207.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx207.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx207.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.30" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx207.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.32" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx207.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx207.34" class="ltx_text ltx_font_typewriter">{133</span><span id="lstnumberx207.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.36" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx207.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.38" class="ltx_text ltx_font_typewriter">1}{2}</span><span id="lstnumberx207.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.40" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx207.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx207.42" class="ltx_text ltx_font_typewriter">66</span><span id="lstnumberx207.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx207.44" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx208" class="ltx_listingline">
<span id="lstnumberx208.1" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx208.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">For</span><span id="lstnumberx208.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.5" class="ltx_text ltx_font_typewriter">(19,</span><span id="lstnumberx208.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.7" class="ltx_text ltx_font_typewriter">7),</span><span id="lstnumberx208.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x</span><span id="lstnumberx208.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.11" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx208.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.13" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx208.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx208.15" class="ltx_text ltx_font_typewriter">{19</span><span id="lstnumberx208.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.17" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx208.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.19" class="ltx_text ltx_font_typewriter">7}{2}</span><span id="lstnumberx208.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.21" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx208.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.23" class="ltx_text ltx_font_typewriter">13</span><span id="lstnumberx208.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx208.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx208.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx208.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.30" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx208.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.32" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx208.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx208.34" class="ltx_text ltx_font_typewriter">{19</span><span id="lstnumberx208.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.36" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx208.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.38" class="ltx_text ltx_font_typewriter">7}{2}</span><span id="lstnumberx208.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.40" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx208.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx208.42" class="ltx_text ltx_font_typewriter">6</span><span id="lstnumberx208.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx208.44" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx209" class="ltx_listingline">
</div>
<div id="lstnumberx210" class="ltx_listingline">
<span id="lstnumberx210.1" class="ltx_text ltx_font_typewriter">(13,</span><span id="lstnumberx210.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx210.3" class="ltx_text ltx_font_typewriter">6)</span><span id="lstnumberx210.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx210.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx210.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx210.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx210.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx210.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx210.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx210.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">smallest</span><span id="lstnumberx210.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx210.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx210.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx210.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx210.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx210.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx210.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx210.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">two</span><span id="lstnumberx210.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx210.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">perfect</span><span id="lstnumberx210.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx210.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">squares</span><span id="lstnumberx210.24" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx211" class="ltx_listingline">
</div>
<div id="lstnumberx212" class="ltx_listingline">
<span id="lstnumberx212.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">So</span><span id="lstnumberx212.2" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx212.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx212.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">let</span><span id="lstnumberx212.5" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx212.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx212.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx212.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">modify</span><span id="lstnumberx212.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx212.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx212.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx212.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">previous</span><span id="lstnumberx212.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx212.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solution</span><span id="lstnumberx212.15" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx213" class="ltx_listingline">
</div>
<div id="lstnumberx214" class="ltx_listingline">
<span id="lstnumberx214.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx214.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span>
</div>
<div id="lstnumberx215" class="ltx_listingline">
<span id="lstnumberx215.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">def</span><span id="lstnumberx215.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx215.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">min_sum_perfect_squares</span><span id="lstnumberx215.4" class="ltx_text ltx_font_typewriter">():</span>
</div>
<div id="lstnumberx216" class="ltx_listingline">
</div>
<div id="lstnumberx217" class="ltx_listingline">
<span id="lstnumberx217.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx217.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">min_squares</span><span id="lstnumberx217.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx217.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx217.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx217.6" class="ltx_text ltx_font_typewriter">[13**2,</span><span id="lstnumberx217.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx217.8" class="ltx_text ltx_font_typewriter">6**2]</span>
</div>
<div id="lstnumberx218" class="ltx_listingline">
<span id="lstnumberx218.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx218.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">min_sum_squares</span><span id="lstnumberx218.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx218.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx218.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx218.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx218.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx218.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">min_squares</span><span id="lstnumberx218.9" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx219" class="ltx_listingline">
</div>
<div id="lstnumberx220" class="ltx_listingline">
<span id="lstnumberx220.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx220.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx220.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx220.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">min_sum_squares</span>
</div>
<div id="lstnumberx221" class="ltx_listingline">
<span id="lstnumberx221.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx222" class="ltx_listingline">
<span id="lstnumberx222.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx222.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span>
</div>
<div id="lstnumberx223" class="ltx_listingline">
<span id="lstnumberx223.1" class="ltx_text ltx_font_typewriter">205</span>
</div>
<div id="lstnumberx224" class="ltx_listingline">
<span id="lstnumberx224.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx225" class="ltx_listingline">
</div>
<div id="lstnumberx226" class="ltx_listingline">
<span id="lstnumberx226.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx226.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx226.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">smallest</span><span id="lstnumberx226.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx226.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">possible</span><span id="lstnumberx226.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx226.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx226.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx226.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx226.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx226.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx226.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx226.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">two</span><span id="lstnumberx226.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx226.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">perfect</span><span id="lstnumberx226.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx226.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">squares</span><span id="lstnumberx226.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx226.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx226.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx226.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx226.22" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx226.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx226.24" class="ltx_text ltx_font_typewriter">{205}</span><span id="lstnumberx226.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx226.26" class="ltx_text ltx_font_typewriter">.</span>
</div>
</div>
</figure>
<figure id="LST4" class="ltx_float ltx_lstlisting">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float">Listing 4: </span>Success case for <span id="LST4.3.1" class="ltx_text ltx_font_smallcaps">ToRA</span>: <span id="LST4.4.2" class="ltx_text ltx_font_bold">Finalizing answer</span>.</figcaption>
<div id="LST4.5" class="ltx_listing ltx_lstlisting ltx_framed ltx_framed_topbottom ltx_listing">
<div class="ltx_listing_data"><a href="data:text/plain;base64,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" download="section/cases/success/finalize.md">⬇</a></div>
<div id="lstnumberx227" class="ltx_listingline">
<span id="lstnumberx227.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Question</span><span id="lstnumberx227.2" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx228" class="ltx_listingline">
<span id="lstnumberx228.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Compute</span><span id="lstnumberx228.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx228.3" class="ltx_text ltx_font_typewriter">\[\</span><span id="lstnumberx228.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_</span><span id="lstnumberx228.5" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx228.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx228.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx228.8" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx228.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx228.10" class="ltx_text ltx_font_typewriter">1}^\</span><span id="lstnumberx228.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">infty</span><span id="lstnumberx228.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx228.13" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx228.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx228.15" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx228.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx228.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx228.18" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx228.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx228.20" class="ltx_text ltx_font_typewriter">2}{</span><span id="lstnumberx228.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx228.22" class="ltx_text ltx_font_typewriter">!</span><span id="lstnumberx228.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx228.24" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx228.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx228.26" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx228.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx228.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx228.29" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx228.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx228.31" class="ltx_text ltx_font_typewriter">1)!</span><span id="lstnumberx228.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx228.33" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx228.34" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx228.35" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx228.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx228.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx228.38" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx228.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx228.40" class="ltx_text ltx_font_typewriter">2)!}.\]</span>
</div>
<div id="lstnumberx229" class="ltx_listingline">
</div>
<div id="lstnumberx230" class="ltx_listingline">
<span id="lstnumberx230.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solution</span><span id="lstnumberx230.2" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx231" class="ltx_listingline">
<span id="lstnumberx231.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx231.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span>
</div>
<div id="lstnumberx232" class="ltx_listingline">
<span id="lstnumberx232.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx232.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx232.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx232.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx232.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx232.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx232.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span><span id="lstnumberx232.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx232.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx232.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Sum</span><span id="lstnumberx232.11" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx232.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx232.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">factorial</span><span id="lstnumberx232.14" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx232.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx232.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">oo</span><span id="lstnumberx232.17" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx232.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx232.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">simplify</span>
</div>
<div id="lstnumberx233" class="ltx_listingline">
</div>
<div id="lstnumberx234" class="ltx_listingline">
<span id="lstnumberx234.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">def</span><span id="lstnumberx234.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx234.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">infinite_sum</span><span id="lstnumberx234.4" class="ltx_text ltx_font_typewriter">():</span>
</div>
<div id="lstnumberx235" class="ltx_listingline">
<span id="lstnumberx235.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx235.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">expr</span><span id="lstnumberx235.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx235.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx235.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx235.6" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx235.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx235.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx235.9" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx235.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx235.11" class="ltx_text ltx_font_typewriter">2)</span><span id="lstnumberx235.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx235.13" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx235.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx235.15" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx235.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">factorial</span><span id="lstnumberx235.17" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx235.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx235.19" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx235.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx235.21" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx235.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx235.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">factorial</span><span id="lstnumberx235.24" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx235.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx235.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx235.27" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx235.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx235.29" class="ltx_text ltx_font_typewriter">1)</span><span id="lstnumberx235.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx235.31" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx235.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx235.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">factorial</span><span id="lstnumberx235.34" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx235.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx235.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx235.37" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx235.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx235.39" class="ltx_text ltx_font_typewriter">2))</span>
</div>
<div id="lstnumberx236" class="ltx_listingline">
<span id="lstnumberx236.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx236.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_expr</span><span id="lstnumberx236.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx236.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx236.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx236.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Sum</span><span id="lstnumberx236.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx236.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">expr</span><span id="lstnumberx236.9" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx236.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx236.11" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx236.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx236.13" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx236.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx236.15" class="ltx_text ltx_font_typewriter">1,</span><span id="lstnumberx236.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx236.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">oo</span><span id="lstnumberx236.18" class="ltx_text ltx_font_typewriter">))</span>
</div>
<div id="lstnumberx237" class="ltx_listingline">
<span id="lstnumberx237.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx237.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx237.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx237.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx237.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx237.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_expr</span><span id="lstnumberx237.7" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx237.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">doit</span><span id="lstnumberx237.9" class="ltx_text ltx_font_typewriter">()</span>
</div>
<div id="lstnumberx238" class="ltx_listingline">
</div>
<div id="lstnumberx239" class="ltx_listingline">
<span id="lstnumberx239.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx239.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx239.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx239.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span>
</div>
<div id="lstnumberx240" class="ltx_listingline">
</div>
<div id="lstnumberx241" class="ltx_listingline">
<span id="lstnumberx241.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx241.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx241.3" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx241.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx241.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">infinite_sum</span><span id="lstnumberx241.6" class="ltx_text ltx_font_typewriter">()</span>
</div>
<div id="lstnumberx242" class="ltx_listingline">
<span id="lstnumberx242.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">print</span><span id="lstnumberx242.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx242.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx242.4" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx243" class="ltx_listingline">
<span id="lstnumberx243.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx244" class="ltx_listingline">
<span id="lstnumberx244.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx244.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span>
</div>
<div id="lstnumberx245" class="ltx_listingline">
<span id="lstnumberx245.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Sum</span><span id="lstnumberx245.2" class="ltx_text ltx_font_typewriter">((</span><span id="lstnumberx245.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx245.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx245.5" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx245.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx245.7" class="ltx_text ltx_font_typewriter">2)/(</span><span id="lstnumberx245.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">factorial</span><span id="lstnumberx245.9" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx245.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx245.11" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx245.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx245.13" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx245.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx245.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">factorial</span><span id="lstnumberx245.16" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx245.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx245.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx245.19" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx245.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx245.21" class="ltx_text ltx_font_typewriter">1)</span><span id="lstnumberx245.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx245.23" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx245.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx245.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">factorial</span><span id="lstnumberx245.26" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx245.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx245.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx245.29" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx245.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx245.31" class="ltx_text ltx_font_typewriter">2)),</span><span id="lstnumberx245.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx245.33" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx245.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx245.35" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx245.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx245.37" class="ltx_text ltx_font_typewriter">1,</span><span id="lstnumberx245.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx245.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">oo</span><span id="lstnumberx245.40" class="ltx_text ltx_font_typewriter">))</span>
</div>
<div id="lstnumberx246" class="ltx_listingline">
<span id="lstnumberx246.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx247" class="ltx_listingline">
<span id="lstnumberx247.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx247.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx247.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">does</span><span id="lstnumberx247.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">not</span><span id="lstnumberx247.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">converge</span><span id="lstnumberx247.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx247.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx247.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">simple</span><span id="lstnumberx247.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">expression</span><span id="lstnumberx247.18" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx247.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">so</span><span id="lstnumberx247.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx247.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">answer</span><span id="lstnumberx247.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx247.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx247.29" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx247.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx247.31" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx247.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_</span><span id="lstnumberx247.33" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx247.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx247.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.36" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx247.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.38" class="ltx_text ltx_font_typewriter">1}^\</span><span id="lstnumberx247.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">infty</span><span id="lstnumberx247.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.41" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx247.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx247.43" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx247.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx247.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.46" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx247.47" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.48" class="ltx_text ltx_font_typewriter">2}{</span><span id="lstnumberx247.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx247.50" class="ltx_text ltx_font_typewriter">!</span><span id="lstnumberx247.51" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.52" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx247.53" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.54" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx247.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx247.56" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.57" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx247.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.59" class="ltx_text ltx_font_typewriter">1)!</span><span id="lstnumberx247.60" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.61" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx247.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.63" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx247.64" class="ltx_text ltx_lst_identifier ltx_font_typewriter">k</span><span id="lstnumberx247.65" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.66" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx247.67" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx247.68" class="ltx_text ltx_font_typewriter">2)!}}</span><span id="lstnumberx247.69" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx247.70" class="ltx_text ltx_font_typewriter">.</span>
</div>
</div>
</figure>
</section>
<section id="A6.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">F.2 </span>Failure Cases</h3>

<figure id="LST5" class="ltx_float ltx_lstlisting">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float">Listing 5: </span>Failure case for <span id="LST5.3.1" class="ltx_text ltx_font_smallcaps">ToRA</span>: <span id="LST5.4.2" class="ltx_text ltx_font_bold">Reasoning Error</span>.</figcaption>
<div id="LST5.5" class="ltx_listing ltx_lstlisting ltx_framed ltx_framed_topbottom ltx_listing">
<div class="ltx_listing_data"><a href="data:text/plain;base64,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" download="section/cases/failure/1_reasoning_error.md">⬇</a></div>
<div id="lstnumberx248" class="ltx_listingline">
<span id="lstnumberx248.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Idx</span><span id="lstnumberx248.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx248.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx248.4" class="ltx_text ltx_font_typewriter">4289</span>
</div>
<div id="lstnumberx249" class="ltx_listingline">
<span id="lstnumberx249.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Type</span><span id="lstnumberx249.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx249.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx249.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Prealgebra</span>
</div>
<div id="lstnumberx250" class="ltx_listingline">
<span id="lstnumberx250.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Level</span><span id="lstnumberx250.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx250.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx250.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Level</span><span id="lstnumberx250.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx250.6" class="ltx_text ltx_font_typewriter">2</span>
</div>
<div id="lstnumberx251" class="ltx_listingline">
<span id="lstnumberx251.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Question</span><span id="lstnumberx251.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx251.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx251.4" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx251.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">What</span><span id="lstnumberx251.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx251.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx251.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx251.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx251.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx251.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">smallest</span><span id="lstnumberx251.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx251.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">positive</span><span id="lstnumberx251.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx251.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">multiple</span><span id="lstnumberx251.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx251.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx251.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx251.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$13$</span><span id="lstnumberx251.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx251.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx251.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx251.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx251.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx251.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">greater</span><span id="lstnumberx251.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx251.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">than</span><span id="lstnumberx251.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx251.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$500</span><span id="lstnumberx251.30" class="ltx_text ltx_font_typewriter">?</span><span id="lstnumberx251.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx251.32" class="ltx_text ltx_font_typewriter">’</span>
</div>
<div id="lstnumberx252" class="ltx_listingline">
<span id="lstnumberx252.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solution</span><span id="lstnumberx252.2" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx253" class="ltx_listingline">
<span id="lstnumberx253.1" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx253.2" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx253.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span>
</div>
<div id="lstnumberx254" class="ltx_listingline">
<span id="lstnumberx254.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">def</span><span id="lstnumberx254.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx254.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">smallest_positive_multiple</span><span id="lstnumberx254.4" class="ltx_text ltx_font_typewriter">():</span>
</div>
<div id="lstnumberx255" class="ltx_listingline">
<span id="lstnumberx255.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx255.2" class="ltx_text ltx_font_typewriter">"""</span><span id="lstnumberx255.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">What</span><span id="lstnumberx255.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx255.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx255.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx255.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx255.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx255.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">smallest</span><span id="lstnumberx255.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx255.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">positive</span><span id="lstnumberx255.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx255.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">multiple</span><span id="lstnumberx255.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx255.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx255.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx255.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$13$</span><span id="lstnumberx255.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx255.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx255.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx255.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx255.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx255.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">greater</span><span id="lstnumberx255.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx255.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">than</span><span id="lstnumberx255.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx255.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$500</span><span id="lstnumberx255.28" class="ltx_text ltx_font_typewriter">?</span><span id="lstnumberx255.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx255.30" class="ltx_text ltx_font_typewriter">"""</span>
</div>
<div id="lstnumberx256" class="ltx_listingline">
<span id="lstnumberx256.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx256.2" class="ltx_text ltx_font_typewriter" style="background-color:#F47874;"># Start from 500 and keep adding 13 until we get a number that is greater than 500</span>
</div>
<div id="lstnumberx257" class="ltx_listingline">
<span id="lstnumberx257.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx257.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">num</span><span id="lstnumberx257.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx257.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx257.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx257.6" class="ltx_text ltx_font_typewriter">500</span>
</div>
<div id="lstnumberx258" class="ltx_listingline">
<span id="lstnumberx258.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx258.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">while</span><span id="lstnumberx258.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx258.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">num</span><span id="lstnumberx258.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx258.6" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx258.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx258.8" class="ltx_text ltx_font_typewriter">500:</span>
</div>
<div id="lstnumberx259" class="ltx_listingline">
<span id="lstnumberx259.1" class="ltx_text ltx_lst_space ltx_font_typewriter">        </span><span id="lstnumberx259.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">num</span><span id="lstnumberx259.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx259.4" class="ltx_text ltx_font_typewriter">+=</span><span id="lstnumberx259.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx259.6" class="ltx_text ltx_font_typewriter">13</span>
</div>
<div id="lstnumberx260" class="ltx_listingline">
</div>
<div id="lstnumberx261" class="ltx_listingline">
<span id="lstnumberx261.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx261.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx261.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx261.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">num</span>
</div>
<div id="lstnumberx262" class="ltx_listingline">
</div>
<div id="lstnumberx263" class="ltx_listingline">
<span id="lstnumberx263.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx263.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx263.3" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx263.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx263.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">smallest_positive_multiple</span><span id="lstnumberx263.6" class="ltx_text ltx_font_typewriter">()</span>
</div>
<div id="lstnumberx264" class="ltx_listingline">
<span id="lstnumberx264.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">print</span><span id="lstnumberx264.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx264.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx264.4" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx265" class="ltx_listingline">
<span id="lstnumberx265.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx266" class="ltx_listingline">
<span id="lstnumberx266.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx266.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span>
</div>
<div id="lstnumberx267" class="ltx_listingline">
<span id="lstnumberx267.1" class="ltx_text ltx_font_typewriter">513</span>
</div>
<div id="lstnumberx268" class="ltx_listingline">
<span id="lstnumberx268.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx269" class="ltx_listingline">
<span id="lstnumberx269.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx269.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx269.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">smallest</span><span id="lstnumberx269.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx269.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">positive</span><span id="lstnumberx269.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx269.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">multiple</span><span id="lstnumberx269.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx269.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx269.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx269.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$13$</span><span id="lstnumberx269.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx269.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx269.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx269.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx269.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx269.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">greater</span><span id="lstnumberx269.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx269.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">than</span><span id="lstnumberx269.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx269.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$500$</span><span id="lstnumberx269.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx269.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx269.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx269.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx269.26" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx269.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx269.28" class="ltx_text ltx_font_typewriter">{513}</span><span id="lstnumberx269.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx269.30" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx270" class="ltx_listingline">
</div>
<div id="lstnumberx271" class="ltx_listingline">
<span id="lstnumberx271.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Execution</span><span id="lstnumberx271.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx271.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx271.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">None</span>
</div>
<div id="lstnumberx272" class="ltx_listingline">
<span id="lstnumberx272.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Prediction</span><span id="lstnumberx272.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx272.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx272.4" class="ltx_text ltx_font_typewriter">’513’</span>
</div>
<div id="lstnumberx273" class="ltx_listingline">
<span id="lstnumberx273.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Ground</span><span id="lstnumberx273.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx273.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Truth</span><span id="lstnumberx273.4" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx273.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx273.6" class="ltx_text ltx_font_typewriter">’507’</span>
</div>
<div id="lstnumberx274" class="ltx_listingline">
<span id="lstnumberx274.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Score</span><span id="lstnumberx274.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx274.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx274.4" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx274.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">False</span><span id="lstnumberx274.6" class="ltx_text ltx_font_typewriter">]</span>
</div>
<div id="lstnumberx275" class="ltx_listingline">
<span id="lstnumberx275.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Reference</span><span id="lstnumberx275.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">CoT</span><span id="lstnumberx275.4" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx275.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.6" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx275.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Dividing</span><span id="lstnumberx275.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$500$</span><span id="lstnumberx275.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">by</span><span id="lstnumberx275.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$13$</span><span id="lstnumberx275.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gives</span><span id="lstnumberx275.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx275.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">quotient</span><span id="lstnumberx275.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$38$</span><span id="lstnumberx275.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">with</span><span id="lstnumberx275.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx275.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">remainder</span><span id="lstnumberx275.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx275.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$6$</span><span id="lstnumberx275.32" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx275.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">In</span><span id="lstnumberx275.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">other</span><span id="lstnumberx275.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">words</span><span id="lstnumberx275.39" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx275.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.41" class="ltx_text ltx_font_typewriter">\\[500</span><span id="lstnumberx275.42" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.43" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx275.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">div</span><span id="lstnumberx275.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.46" class="ltx_text ltx_font_typewriter">13=13</span><span id="lstnumberx275.47" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.48" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx275.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cdot</span><span id="lstnumberx275.50" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.51" class="ltx_text ltx_font_typewriter">38+6.\\]</span><span id="lstnumberx275.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">So</span><span id="lstnumberx275.53" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx275.54" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx275.56" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">smallest</span><span id="lstnumberx275.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">positive</span><span id="lstnumberx275.60" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">multiple</span><span id="lstnumberx275.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx275.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.65" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$13$</span><span id="lstnumberx275.66" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx275.68" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.69" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx275.70" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.71" class="ltx_text ltx_lst_identifier ltx_font_typewriter">greater</span><span id="lstnumberx275.72" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.73" class="ltx_text ltx_lst_identifier ltx_font_typewriter">than</span><span id="lstnumberx275.74" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.75" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$500$</span><span id="lstnumberx275.76" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.77" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx275.78" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx275.79" class="ltx_text ltx_font_typewriter">\\[13\\</span><span id="lstnumberx275.80" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cdot39</span><span id="lstnumberx275.81" class="ltx_text ltx_font_typewriter">=\\</span><span id="lstnumberx275.82" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx275.83" class="ltx_text ltx_font_typewriter">{507}.\\]’</span>
</div>
</div>
</figure>
<figure id="LST6" class="ltx_float ltx_lstlisting">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float">Listing 6: </span>Failure case for <span id="LST6.3.1" class="ltx_text ltx_font_smallcaps">ToRA</span>: <span id="LST6.4.2" class="ltx_text ltx_font_bold">Hallucination</span>.</figcaption>
<div id="LST6.5" class="ltx_listing ltx_lstlisting ltx_framed ltx_framed_topbottom ltx_listing">
<div class="ltx_listing_data"><a href="data:text/plain;base64,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" download="section/cases/failure/2_hallucination.md">⬇</a></div>
<div id="lstnumberx276" class="ltx_listingline">
<span id="lstnumberx276.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Idx</span><span id="lstnumberx276.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx276.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx276.4" class="ltx_text ltx_font_typewriter">2775</span>
</div>
<div id="lstnumberx277" class="ltx_listingline">
<span id="lstnumberx277.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Type</span><span id="lstnumberx277.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx277.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx277.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Intermediate</span><span id="lstnumberx277.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx277.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Algebra</span>
</div>
<div id="lstnumberx278" class="ltx_listingline">
<span id="lstnumberx278.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Level</span><span id="lstnumberx278.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx278.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx278.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Level</span><span id="lstnumberx278.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx278.6" class="ltx_text ltx_font_typewriter">3</span>
</div>
<div id="lstnumberx279" class="ltx_listingline">
<span id="lstnumberx279.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Question</span><span id="lstnumberx279.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx279.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.4" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx279.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Find</span><span id="lstnumberx279.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx279.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">eighth</span><span id="lstnumberx279.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">term</span><span id="lstnumberx279.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx279.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx279.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sequence</span><span id="lstnumberx279.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$1440</span><span id="lstnumberx279.20" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx279.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.22" class="ltx_text ltx_font_typewriter">1716,</span><span id="lstnumberx279.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.24" class="ltx_text ltx_font_typewriter">1848,</span><span id="lstnumberx279.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.26" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx279.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ldots$</span><span id="lstnumberx279.28" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx279.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">whose</span><span id="lstnumberx279.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">terms</span><span id="lstnumberx279.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx279.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">formed</span><span id="lstnumberx279.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">by</span><span id="lstnumberx279.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">multiplying</span><span id="lstnumberx279.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx279.43" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">corresponding</span><span id="lstnumberx279.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">terms</span><span id="lstnumberx279.47" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.48" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx279.49" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.50" class="ltx_text ltx_lst_identifier ltx_font_typewriter">two</span><span id="lstnumberx279.51" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">arithmetic</span><span id="lstnumberx279.53" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx279.54" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sequences</span><span id="lstnumberx279.55" class="ltx_text ltx_font_typewriter">.’</span>
</div>
<div id="lstnumberx280" class="ltx_listingline">
<span id="lstnumberx280.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solution</span><span id="lstnumberx280.2" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx281" class="ltx_listingline">
<span id="lstnumberx281.1" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx281.2" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx281.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span>
</div>
<div id="lstnumberx282" class="ltx_listingline">
<span id="lstnumberx282.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx282.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx282.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx282.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx282.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx282.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx282.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span><span id="lstnumberx282.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx282.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx282.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Rational</span><span id="lstnumberx282.11" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx282.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx282.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">simplify</span>
</div>
<div id="lstnumberx283" class="ltx_listingline">
</div>
<div id="lstnumberx284" class="ltx_listingline">
<span id="lstnumberx284.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">def</span><span id="lstnumberx284.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx284.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">eighth_term</span><span id="lstnumberx284.4" class="ltx_text ltx_font_typewriter">():</span>
</div>
<div id="lstnumberx285" class="ltx_listingline">
<span id="lstnumberx285.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx285.2" class="ltx_text ltx_font_typewriter">"""</span><span id="lstnumberx285.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Find</span><span id="lstnumberx285.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx285.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">eighth</span><span id="lstnumberx285.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">term</span><span id="lstnumberx285.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx285.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx285.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sequence</span><span id="lstnumberx285.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$1440</span><span id="lstnumberx285.18" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx285.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.20" class="ltx_text ltx_font_typewriter">1716,</span><span id="lstnumberx285.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.22" class="ltx_text ltx_font_typewriter">1848,</span><span id="lstnumberx285.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.24" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx285.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ldots$</span><span id="lstnumberx285.26" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx285.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">whose</span><span id="lstnumberx285.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">terms</span><span id="lstnumberx285.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx285.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">formed</span><span id="lstnumberx285.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">by</span><span id="lstnumberx285.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">multiplying</span><span id="lstnumberx285.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx285.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">corresponding</span><span id="lstnumberx285.43" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">terms</span><span id="lstnumberx285.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx285.47" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.48" class="ltx_text ltx_lst_identifier ltx_font_typewriter">two</span><span id="lstnumberx285.49" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.50" class="ltx_text ltx_lst_identifier ltx_font_typewriter">arithmetic</span><span id="lstnumberx285.51" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx285.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sequences</span><span id="lstnumberx285.53" class="ltx_text ltx_font_typewriter">."""</span>
</div>
<div id="lstnumberx286" class="ltx_listingline">
<span id="lstnumberx286.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx286.2" class="ltx_text ltx_font_typewriter" style="background-color:#F47874;">a1, d1 = 1440, 36</span>
</div>
<div id="lstnumberx287" class="ltx_listingline">
<span id="lstnumberx287.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx287.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a2</span><span id="lstnumberx287.3" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx287.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx287.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">d2</span><span id="lstnumberx287.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx287.7" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx287.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx287.9" class="ltx_text ltx_font_typewriter">16,</span><span id="lstnumberx287.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx287.11" class="ltx_text ltx_font_typewriter">2</span>
</div>
<div id="lstnumberx288" class="ltx_listingline">
</div>
<div id="lstnumberx289" class="ltx_listingline">
<span id="lstnumberx289.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx289.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx289.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx289.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Calculate</span><span id="lstnumberx289.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx289.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx289.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx289.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">eighth</span><span id="lstnumberx289.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx289.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">term</span><span id="lstnumberx289.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx289.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx289.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx289.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">each</span><span id="lstnumberx289.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx289.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sequence</span>
</div>
<div id="lstnumberx290" class="ltx_listingline">
<span id="lstnumberx290.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx290.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">term1</span><span id="lstnumberx290.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx290.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx290.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx290.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a1</span><span id="lstnumberx290.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx290.8" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx290.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx290.10" class="ltx_text ltx_font_typewriter">7*</span><span id="lstnumberx290.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">d1</span>
</div>
<div id="lstnumberx291" class="ltx_listingline">
<span id="lstnumberx291.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx291.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">term2</span><span id="lstnumberx291.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx291.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx291.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx291.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a2</span><span id="lstnumberx291.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx291.8" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx291.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx291.10" class="ltx_text ltx_font_typewriter">7*</span><span id="lstnumberx291.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">d2</span>
</div>
<div id="lstnumberx292" class="ltx_listingline">
</div>
<div id="lstnumberx293" class="ltx_listingline">
<span id="lstnumberx293.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx293.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx293.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx293.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Multiply</span><span id="lstnumberx293.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx293.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx293.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx293.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">terms</span>
</div>
<div id="lstnumberx294" class="ltx_listingline">
<span id="lstnumberx294.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx294.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx294.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx294.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx294.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx294.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">term1</span><span id="lstnumberx294.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx294.8" class="ltx_text ltx_font_typewriter">*</span><span id="lstnumberx294.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx294.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">term2</span>
</div>
<div id="lstnumberx295" class="ltx_listingline">
</div>
<div id="lstnumberx296" class="ltx_listingline">
<span id="lstnumberx296.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx296.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx296.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx296.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span>
</div>
<div id="lstnumberx297" class="ltx_listingline">
</div>
<div id="lstnumberx298" class="ltx_listingline">
<span id="lstnumberx298.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx298.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx298.3" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx298.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx298.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">eighth_term</span><span id="lstnumberx298.6" class="ltx_text ltx_font_typewriter">()</span>
</div>
<div id="lstnumberx299" class="ltx_listingline">
<span id="lstnumberx299.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">print</span><span id="lstnumberx299.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx299.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx299.4" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx300" class="ltx_listingline">
<span id="lstnumberx300.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx301" class="ltx_listingline">
<span id="lstnumberx301.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx301.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span>
</div>
<div id="lstnumberx302" class="ltx_listingline">
<span id="lstnumberx302.1" class="ltx_text ltx_font_typewriter">50760</span>
</div>
<div id="lstnumberx303" class="ltx_listingline">
<span id="lstnumberx303.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx304" class="ltx_listingline">
<span id="lstnumberx304.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx304.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx304.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">eighth</span><span id="lstnumberx304.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx304.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">term</span><span id="lstnumberx304.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx304.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx304.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx304.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx304.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx304.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sequence</span><span id="lstnumberx304.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx304.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx304.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx304.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx304.16" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx304.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx304.18" class="ltx_text ltx_font_typewriter">{50760}</span><span id="lstnumberx304.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx304.20" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx305" class="ltx_listingline">
</div>
<div id="lstnumberx306" class="ltx_listingline">
<span id="lstnumberx306.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Execution</span><span id="lstnumberx306.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx306.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx306.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">None</span>
</div>
<div id="lstnumberx307" class="ltx_listingline">
<span id="lstnumberx307.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Prediction</span><span id="lstnumberx307.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx307.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx307.4" class="ltx_text ltx_font_typewriter">’50760’</span>
</div>
<div id="lstnumberx308" class="ltx_listingline">
<span id="lstnumberx308.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Ground</span><span id="lstnumberx308.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx308.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Truth</span><span id="lstnumberx308.4" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx308.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx308.6" class="ltx_text ltx_font_typewriter">’348’</span>
</div>
<div id="lstnumberx309" class="ltx_listingline">
<span id="lstnumberx309.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Score</span><span id="lstnumberx309.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx309.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx309.4" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx309.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">False</span><span id="lstnumberx309.6" class="ltx_text ltx_font_typewriter">]</span>
</div>
<div id="lstnumberx310" class="ltx_listingline">
<span id="lstnumberx310.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Reference</span><span id="lstnumberx310.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">CoT</span><span id="lstnumberx310.4" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx310.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.6" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx310.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">If</span><span id="lstnumberx310.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$a_n</span><span id="lstnumberx310.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.11" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx310.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">an</span><span id="lstnumberx310.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.14" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b$</span><span id="lstnumberx310.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx310.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$b_n</span><span id="lstnumberx310.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.22" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx310.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cn</span><span id="lstnumberx310.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.26" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">d$</span><span id="lstnumberx310.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx310.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">two</span><span id="lstnumberx310.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">arithmetic</span><span id="lstnumberx310.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sequences</span><span id="lstnumberx310.37" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx310.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">then</span><span id="lstnumberx310.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">their</span><span id="lstnumberx310.42" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">term</span><span id="lstnumberx310.44" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx310.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">wise</span><span id="lstnumberx310.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">product</span><span id="lstnumberx310.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">takes</span><span id="lstnumberx310.50" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx310.52" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.53" class="ltx_text ltx_lst_identifier ltx_font_typewriter">form</span><span id="lstnumberx310.54" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.55" class="ltx_text ltx_font_typewriter">\\[</span><span id="lstnumberx310.56" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a_nb_n</span><span id="lstnumberx310.57" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.58" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx310.59" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.60" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx310.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">an</span><span id="lstnumberx310.62" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b</span><span id="lstnumberx310.64" class="ltx_text ltx_font_typewriter">)(</span><span id="lstnumberx310.65" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cn</span><span id="lstnumberx310.66" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">d</span><span id="lstnumberx310.68" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx310.69" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.70" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx310.71" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.72" class="ltx_text ltx_lst_identifier ltx_font_typewriter">An</span><span id="lstnumberx310.73" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx310.74" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.75" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.76" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.77" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Bn</span><span id="lstnumberx310.78" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.79" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.80" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.81" class="ltx_text ltx_lst_identifier ltx_font_typewriter">C</span><span id="lstnumberx310.82" class="ltx_text ltx_font_typewriter">,\\]</span><span id="lstnumberx310.83" class="ltx_text ltx_lst_identifier ltx_font_typewriter">where</span><span id="lstnumberx310.84" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.85" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$A</span><span id="lstnumberx310.86" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx310.87" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.88" class="ltx_text ltx_lst_identifier ltx_font_typewriter">B</span><span id="lstnumberx310.89" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx310.90" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.91" class="ltx_text ltx_lst_identifier ltx_font_typewriter">C$</span><span id="lstnumberx310.92" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.93" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx310.94" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.95" class="ltx_text ltx_lst_identifier ltx_font_typewriter">constants</span><span id="lstnumberx310.96" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx310.97" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.98" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Therefore</span><span id="lstnumberx310.99" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx310.100" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.101" class="ltx_text ltx_lst_identifier ltx_font_typewriter">some</span><span id="lstnumberx310.102" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.103" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sequence</span><span id="lstnumberx310.104" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.105" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x_n</span><span id="lstnumberx310.106" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.107" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx310.108" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.109" class="ltx_text ltx_lst_identifier ltx_font_typewriter">An</span><span id="lstnumberx310.110" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx310.111" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.112" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.113" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.114" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Bn</span><span id="lstnumberx310.115" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.116" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.117" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.118" class="ltx_text ltx_lst_identifier ltx_font_typewriter">C$</span><span id="lstnumberx310.119" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.120" class="ltx_text ltx_lst_identifier ltx_font_typewriter">has</span><span id="lstnumberx310.121" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.122" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x_0</span><span id="lstnumberx310.123" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.124" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx310.125" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.126" class="ltx_text ltx_font_typewriter">1440,</span><span id="lstnumberx310.127" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx310.128" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.129" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x_1</span><span id="lstnumberx310.130" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.131" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx310.132" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.133" class="ltx_text ltx_font_typewriter">1716,</span><span id="lstnumberx310.134" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx310.135" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.136" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx310.137" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.138" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x_2</span><span id="lstnumberx310.139" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.140" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx310.141" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.142" class="ltx_text ltx_font_typewriter">1848.</span><span id="lstnumberx310.143" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx310.144" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.145" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Then</span><span id="lstnumberx310.146" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.147" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx310.148" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.149" class="ltx_text ltx_lst_identifier ltx_font_typewriter">have</span><span id="lstnumberx310.150" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.151" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx310.152" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.153" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equations</span><span id="lstnumberx310.154" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.155" class="ltx_text ltx_font_typewriter">\\[\\</span><span id="lstnumberx310.156" class="ltx_text ltx_lst_identifier ltx_font_typewriter">begin</span><span id="lstnumberx310.157" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx310.158" class="ltx_text ltx_lst_identifier ltx_font_typewriter">aligned</span><span id="lstnumberx310.159" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx310.160" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.161" class="ltx_text ltx_lst_identifier ltx_font_typewriter">C</span><span id="lstnumberx310.162" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.163" class="ltx_text ltx_font_typewriter">&amp;=</span><span id="lstnumberx310.164" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.165" class="ltx_text ltx_font_typewriter">1440,</span><span id="lstnumberx310.166" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.167" class="ltx_text ltx_font_typewriter">\\\\</span><span id="lstnumberx310.168" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.169" class="ltx_text ltx_lst_identifier ltx_font_typewriter">A</span><span id="lstnumberx310.170" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.171" class="ltx_text ltx_lst_identifier ltx_font_typewriter">B</span><span id="lstnumberx310.172" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.173" class="ltx_text ltx_lst_identifier ltx_font_typewriter">C</span><span id="lstnumberx310.174" class="ltx_text ltx_font_typewriter">&amp;=1716,\\\\</span><span id="lstnumberx310.175" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx310.176" class="ltx_text ltx_font_typewriter">4</span><span id="lstnumberx310.177" class="ltx_text ltx_lst_identifier ltx_font_typewriter">A</span><span id="lstnumberx310.178" class="ltx_text ltx_font_typewriter">+2</span><span id="lstnumberx310.179" class="ltx_text ltx_lst_identifier ltx_font_typewriter">B</span><span id="lstnumberx310.180" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.181" class="ltx_text ltx_lst_identifier ltx_font_typewriter">C</span><span id="lstnumberx310.182" class="ltx_text ltx_font_typewriter">&amp;=1848.</span><span id="lstnumberx310.183" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.184" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx310.185" class="ltx_text ltx_lst_identifier ltx_font_typewriter">end</span><span id="lstnumberx310.186" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx310.187" class="ltx_text ltx_lst_identifier ltx_font_typewriter">aligned</span><span id="lstnumberx310.188" class="ltx_text ltx_font_typewriter">}\\]</span><span id="lstnumberx310.189" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Subtracting</span><span id="lstnumberx310.190" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.191" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$C</span><span id="lstnumberx310.192" class="ltx_text ltx_font_typewriter">=1440</span><span id="lstnumberx310.193" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx310.194" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.195" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx310.196" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.197" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx310.198" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.199" class="ltx_text ltx_lst_identifier ltx_font_typewriter">second</span><span id="lstnumberx310.200" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.201" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx310.202" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.203" class="ltx_text ltx_lst_identifier ltx_font_typewriter">third</span><span id="lstnumberx310.204" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.205" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equations</span><span id="lstnumberx310.206" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx310.207" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.208" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx310.209" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.210" class="ltx_text ltx_lst_identifier ltx_font_typewriter">have</span><span id="lstnumberx310.211" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.212" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$A</span><span id="lstnumberx310.213" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.214" class="ltx_text ltx_lst_identifier ltx_font_typewriter">B</span><span id="lstnumberx310.215" class="ltx_text ltx_font_typewriter">=276</span><span id="lstnumberx310.216" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx310.217" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.218" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx310.219" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.220" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$4A</span><span id="lstnumberx310.221" class="ltx_text ltx_font_typewriter">+2</span><span id="lstnumberx310.222" class="ltx_text ltx_lst_identifier ltx_font_typewriter">B</span><span id="lstnumberx310.223" class="ltx_text ltx_font_typewriter">=408,</span><span id="lstnumberx310.224" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx310.225" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.226" class="ltx_text ltx_lst_identifier ltx_font_typewriter">or</span><span id="lstnumberx310.227" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.228" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$2A</span><span id="lstnumberx310.229" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.230" class="ltx_text ltx_lst_identifier ltx_font_typewriter">B</span><span id="lstnumberx310.231" class="ltx_text ltx_font_typewriter">=204.</span><span id="lstnumberx310.232" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx310.233" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.234" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Then</span><span id="lstnumberx310.235" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.236" class="ltx_text ltx_font_typewriter">\\[</span><span id="lstnumberx310.237" class="ltx_text ltx_lst_identifier ltx_font_typewriter">A</span><span id="lstnumberx310.238" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.239" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx310.240" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.241" class="ltx_text ltx_font_typewriter">(2</span><span id="lstnumberx310.242" class="ltx_text ltx_lst_identifier ltx_font_typewriter">A</span><span id="lstnumberx310.243" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.244" class="ltx_text ltx_lst_identifier ltx_font_typewriter">B</span><span id="lstnumberx310.245" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx310.246" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.247" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx310.248" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.249" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx310.250" class="ltx_text ltx_lst_identifier ltx_font_typewriter">A</span><span id="lstnumberx310.251" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.252" class="ltx_text ltx_lst_identifier ltx_font_typewriter">B</span><span id="lstnumberx310.253" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx310.254" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.255" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx310.256" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.257" class="ltx_text ltx_font_typewriter">204</span><span id="lstnumberx310.258" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.259" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx310.260" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.261" class="ltx_text ltx_font_typewriter">276</span><span id="lstnumberx310.262" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.263" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx310.264" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.265" class="ltx_text ltx_font_typewriter">-72,\\]</span><span id="lstnumberx310.266" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx310.267" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.268" class="ltx_text ltx_lst_identifier ltx_font_typewriter">so</span><span id="lstnumberx310.269" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.270" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$B</span><span id="lstnumberx310.271" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.272" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx310.273" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.274" class="ltx_text ltx_font_typewriter">276-</span><span id="lstnumberx310.275" class="ltx_text ltx_lst_identifier ltx_font_typewriter">A</span><span id="lstnumberx310.276" class="ltx_text ltx_font_typewriter">=348.</span><span id="lstnumberx310.277" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx310.278" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.279" class="ltx_text ltx_lst_identifier ltx_font_typewriter">We</span><span id="lstnumberx310.280" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.281" class="ltx_text ltx_lst_identifier ltx_font_typewriter">conclude</span><span id="lstnumberx310.282" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.283" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx310.284" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.285" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx310.286" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.287" class="ltx_text ltx_lst_identifier ltx_font_typewriter">given</span><span id="lstnumberx310.288" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.289" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sequence</span><span id="lstnumberx310.290" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.291" class="ltx_text ltx_lst_identifier ltx_font_typewriter">has</span><span id="lstnumberx310.292" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.293" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx310.294" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.295" class="ltx_text ltx_lst_identifier ltx_font_typewriter">general</span><span id="lstnumberx310.296" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.297" class="ltx_text ltx_lst_identifier ltx_font_typewriter">formula</span><span id="lstnumberx310.298" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.299" class="ltx_text ltx_font_typewriter">\\[</span><span id="lstnumberx310.300" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x_n</span><span id="lstnumberx310.301" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.302" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx310.303" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.304" class="ltx_text ltx_font_typewriter">-72</span><span id="lstnumberx310.305" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx310.306" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx310.307" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.308" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.309" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.310" class="ltx_text ltx_font_typewriter">348</span><span id="lstnumberx310.311" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx310.312" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.313" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.314" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.315" class="ltx_text ltx_font_typewriter">1440.\\]</span><span id="lstnumberx310.316" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Then</span><span id="lstnumberx310.317" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.318" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx310.319" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.320" class="ltx_text ltx_lst_identifier ltx_font_typewriter">eighth</span><span id="lstnumberx310.321" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.322" class="ltx_text ltx_lst_identifier ltx_font_typewriter">term</span><span id="lstnumberx310.323" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.324" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx310.325" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.326" class="ltx_text ltx_font_typewriter">\\[\\</span><span id="lstnumberx310.327" class="ltx_text ltx_lst_identifier ltx_font_typewriter">begin</span><span id="lstnumberx310.328" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx310.329" class="ltx_text ltx_lst_identifier ltx_font_typewriter">aligned</span><span id="lstnumberx310.330" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx310.331" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.332" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x_7</span><span id="lstnumberx310.333" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.334" class="ltx_text ltx_font_typewriter">&amp;=</span><span id="lstnumberx310.335" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.336" class="ltx_text ltx_font_typewriter">-72</span><span id="lstnumberx310.337" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.338" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx310.339" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cdot</span><span id="lstnumberx310.340" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.341" class="ltx_text ltx_font_typewriter">7^2</span><span id="lstnumberx310.342" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.343" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.344" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.345" class="ltx_text ltx_font_typewriter">348</span><span id="lstnumberx310.346" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.347" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx310.348" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cdot</span><span id="lstnumberx310.349" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.350" class="ltx_text ltx_font_typewriter">7</span><span id="lstnumberx310.351" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.352" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx310.353" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.354" class="ltx_text ltx_font_typewriter">1440</span><span id="lstnumberx310.355" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.356" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx310.357" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.358" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx310.359" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx310.360" class="ltx_text ltx_font_typewriter">{348}.</span><span id="lstnumberx310.361" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx310.362" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx310.363" class="ltx_text ltx_lst_identifier ltx_font_typewriter">end</span><span id="lstnumberx310.364" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx310.365" class="ltx_text ltx_lst_identifier ltx_font_typewriter">aligned</span><span id="lstnumberx310.366" class="ltx_text ltx_font_typewriter">}\\]’</span>
</div>
</div>
</figure>
<figure id="LST7" class="ltx_float ltx_lstlisting">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float">Listing 7: </span>Failure case for <span id="LST7.4.1" class="ltx_text ltx_font_smallcaps">ToRA</span>: <span id="LST7.5.2" class="ltx_text ltx_font_bold">Diagram Understanding</span>. Navigating the complexities of input diagrams is key to solving many problems in Geometry, Precalculus, and Intermediate Algebra. This ideal process encompasses not just visual modalities, but also involves interactions with diagrams, such as adding auxiliary lines. Nonetheless, the utility of computational tools like <span id="LST7.6.3" class="ltx_text ltx_font_typewriter">SymPy</span> is, unfortunately, only marginally beneficial.</figcaption>
<div id="LST7.7" class="ltx_listing ltx_lstlisting ltx_framed ltx_framed_topbottom ltx_listing">
<div class="ltx_listing_data"><a href="data:text/plain;base64,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" download="section/cases/failure/3_diagram_understanding.md">⬇</a></div>
<div id="lstnumberx311" class="ltx_listingline">
<span id="lstnumberx311.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Question</span><span id="lstnumberx311.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx311.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">A</span><span id="lstnumberx311.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">square</span><span id="lstnumberx311.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">has</span><span id="lstnumberx311.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx311.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">side</span><span id="lstnumberx311.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">length</span><span id="lstnumberx311.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx311.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.18" class="ltx_text ltx_font_typewriter">10</span><span id="lstnumberx311.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">inches</span><span id="lstnumberx311.21" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx311.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Congruent</span><span id="lstnumberx311.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">isosceles</span><span id="lstnumberx311.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">right</span><span id="lstnumberx311.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">triangles</span><span id="lstnumberx311.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx311.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cut</span><span id="lstnumberx311.34" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">off</span><span id="lstnumberx311.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">each</span><span id="lstnumberx311.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">corner</span><span id="lstnumberx311.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">so</span><span id="lstnumberx311.42" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx311.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx311.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">resulting</span><span id="lstnumberx311.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">octagon</span><span id="lstnumberx311.50" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">has</span><span id="lstnumberx311.52" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.53" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equal</span><span id="lstnumberx311.54" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">side</span><span id="lstnumberx311.56" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">lengths</span><span id="lstnumberx311.58" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx311.59" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.60" class="ltx_text ltx_lst_identifier ltx_font_typewriter">How</span><span id="lstnumberx311.61" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.62" class="ltx_text ltx_lst_identifier ltx_font_typewriter">many</span><span id="lstnumberx311.63" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.64" class="ltx_text ltx_lst_identifier ltx_font_typewriter">inches</span><span id="lstnumberx311.65" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.66" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx311.67" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.68" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx311.69" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.70" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx311.71" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.72" class="ltx_text ltx_lst_identifier ltx_font_typewriter">length</span><span id="lstnumberx311.73" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.74" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx311.75" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.76" class="ltx_text ltx_lst_identifier ltx_font_typewriter">one</span><span id="lstnumberx311.77" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.78" class="ltx_text ltx_lst_identifier ltx_font_typewriter">side</span><span id="lstnumberx311.79" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.80" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx311.81" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.82" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx311.83" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.84" class="ltx_text ltx_lst_identifier ltx_font_typewriter">octagon</span><span id="lstnumberx311.85" class="ltx_text ltx_font_typewriter">?</span><span id="lstnumberx311.86" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.87" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Express</span><span id="lstnumberx311.88" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.89" class="ltx_text ltx_lst_identifier ltx_font_typewriter">your</span><span id="lstnumberx311.90" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.91" class="ltx_text ltx_lst_identifier ltx_font_typewriter">answer</span><span id="lstnumberx311.92" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.93" class="ltx_text ltx_lst_identifier ltx_font_typewriter">as</span><span id="lstnumberx311.94" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.95" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx311.96" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.97" class="ltx_text ltx_lst_identifier ltx_font_typewriter">decimal</span><span id="lstnumberx311.98" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.99" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx311.100" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.101" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx311.102" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.103" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nearest</span><span id="lstnumberx311.104" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx311.105" class="ltx_text ltx_lst_identifier ltx_font_typewriter">hundredth</span><span id="lstnumberx311.106" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx312" class="ltx_listingline">
</div>
<div id="lstnumberx313" class="ltx_listingline">
<span id="lstnumberx313.1" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx313.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asy</span><span id="lstnumberx313.3" class="ltx_text ltx_font_typewriter">]</span>
</div>
<div id="lstnumberx314" class="ltx_listingline">
<span id="lstnumberx314.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">size</span><span id="lstnumberx314.2" class="ltx_text ltx_font_typewriter">(150);</span>
</div>
<div id="lstnumberx315" class="ltx_listingline">
<span id="lstnumberx315.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx315.2" class="ltx_text ltx_font_typewriter">((0,0)--(10,0)--(10,10)--(0,10)--(0,0),</span><span id="lstnumberx315.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linewidth</span><span id="lstnumberx315.4" class="ltx_text ltx_font_typewriter">(0.7));</span>
</div>
<div id="lstnumberx316" class="ltx_listingline">
<span id="lstnumberx316.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx316.2" class="ltx_text ltx_font_typewriter">((5*</span><span id="lstnumberx316.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx316.4" class="ltx_text ltx_font_typewriter">(2)/(1</span><span id="lstnumberx316.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx316.6" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx316.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx316.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx316.9" class="ltx_text ltx_font_typewriter">(2)),0)--(0,5*</span><span id="lstnumberx316.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx316.11" class="ltx_text ltx_font_typewriter">(2)/(1</span><span id="lstnumberx316.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx316.13" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx316.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx316.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx316.16" class="ltx_text ltx_font_typewriter">(2))),</span><span id="lstnumberx316.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linewidth</span><span id="lstnumberx316.18" class="ltx_text ltx_font_typewriter">(0.7));</span>
</div>
<div id="lstnumberx317" class="ltx_listingline">
<span id="lstnumberx317.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx317.2" class="ltx_text ltx_font_typewriter">((10</span><span id="lstnumberx317.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx317.4" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx317.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx317.6" class="ltx_text ltx_font_typewriter">(5*</span><span id="lstnumberx317.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx317.8" class="ltx_text ltx_font_typewriter">(2)/(1</span><span id="lstnumberx317.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx317.10" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx317.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx317.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx317.13" class="ltx_text ltx_font_typewriter">(2))),0)--(10,5*</span><span id="lstnumberx317.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx317.15" class="ltx_text ltx_font_typewriter">(2)/(1</span><span id="lstnumberx317.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx317.17" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx317.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx317.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx317.20" class="ltx_text ltx_font_typewriter">(2))),</span><span id="lstnumberx317.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linewidth</span><span id="lstnumberx317.22" class="ltx_text ltx_font_typewriter">(0.7));</span>
</div>
<div id="lstnumberx318" class="ltx_listingline">
<span id="lstnumberx318.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx318.2" class="ltx_text ltx_font_typewriter">((0,(10-(5*</span><span id="lstnumberx318.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx318.4" class="ltx_text ltx_font_typewriter">(2)/(1+</span><span id="lstnumberx318.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx318.6" class="ltx_text ltx_font_typewriter">(2)))))--(5*</span><span id="lstnumberx318.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx318.8" class="ltx_text ltx_font_typewriter">(2)/(1</span><span id="lstnumberx318.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx318.10" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx318.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx318.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx318.13" class="ltx_text ltx_font_typewriter">(2)),10),</span><span id="lstnumberx318.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linewidth</span><span id="lstnumberx318.15" class="ltx_text ltx_font_typewriter">(0.7));</span>
</div>
<div id="lstnumberx319" class="ltx_listingline">
<span id="lstnumberx319.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx319.2" class="ltx_text ltx_font_typewriter">((10</span><span id="lstnumberx319.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx319.4" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx319.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx319.6" class="ltx_text ltx_font_typewriter">(5*</span><span id="lstnumberx319.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx319.8" class="ltx_text ltx_font_typewriter">(2)/(1</span><span id="lstnumberx319.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx319.10" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx319.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx319.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx319.13" class="ltx_text ltx_font_typewriter">(2))),10)--(10,10</span><span id="lstnumberx319.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx319.15" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx319.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx319.17" class="ltx_text ltx_font_typewriter">5*</span><span id="lstnumberx319.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx319.19" class="ltx_text ltx_font_typewriter">(2)/(1</span><span id="lstnumberx319.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx319.21" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx319.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx319.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx319.24" class="ltx_text ltx_font_typewriter">(2))),</span><span id="lstnumberx319.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linewidth</span><span id="lstnumberx319.26" class="ltx_text ltx_font_typewriter">(0.7));</span>
</div>
<div id="lstnumberx320" class="ltx_listingline">
<span id="lstnumberx320.1" class="ltx_text ltx_font_typewriter">[/</span><span id="lstnumberx320.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asy</span><span id="lstnumberx320.3" class="ltx_text ltx_font_typewriter">]</span>
</div>
<div id="lstnumberx321" class="ltx_listingline">
</div>
<div id="lstnumberx322" class="ltx_listingline">
<span id="lstnumberx322.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solution</span><span id="lstnumberx322.2" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx323" class="ltx_listingline">
<span id="lstnumberx323.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx323.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">side</span><span id="lstnumberx323.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">length</span><span id="lstnumberx323.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx323.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx323.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">octagon</span><span id="lstnumberx323.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx323.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx323.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">hypotenuse</span><span id="lstnumberx323.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx323.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx323.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">right</span><span id="lstnumberx323.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">triangle</span><span id="lstnumberx323.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">formed</span><span id="lstnumberx323.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">by</span><span id="lstnumberx323.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx323.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">side</span><span id="lstnumberx323.34" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx323.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx323.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">square</span><span id="lstnumberx323.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx323.42" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx323.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">diagonal</span><span id="lstnumberx323.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx323.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx323.50" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx323.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">octagon</span>
</div>
<div id="lstnumberx324" class="ltx_listingline">
<span id="lstnumberx324.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx324.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">hypotenuse</span><span id="lstnumberx324.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx324.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx324.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx324.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx324.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx324.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">side</span><span id="lstnumberx324.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">length</span><span id="lstnumberx324.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx324.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx324.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">square</span><span id="lstnumberx324.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx324.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx324.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">length</span><span id="lstnumberx324.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx324.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx324.34" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">diagonal</span><span id="lstnumberx324.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx324.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx324.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx324.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">octagon</span>
</div>
<div id="lstnumberx325" class="ltx_listingline">
<span id="lstnumberx325.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx325.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx325.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">length</span><span id="lstnumberx325.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx325.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx325.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx325.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx325.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx325.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">diagonal</span><span id="lstnumberx325.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx325.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx325.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx325.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx325.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx325.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">octagon</span><span id="lstnumberx325.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx325.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx325.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx325.19" class="ltx_text ltx_font_typewriter">5*</span><span id="lstnumberx325.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx325.21" class="ltx_text ltx_font_typewriter">(2)/(1</span><span id="lstnumberx325.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx325.23" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx325.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx325.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx325.26" class="ltx_text ltx_font_typewriter">(2))</span>
</div>
<div id="lstnumberx326" class="ltx_listingline">
</div>
<div id="lstnumberx327" class="ltx_listingline">
<span id="lstnumberx327.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx327.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span>
</div>
<div id="lstnumberx328" class="ltx_listingline">
<span id="lstnumberx328.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx328.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx328.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx328.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx328.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx328.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx328.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx328.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx328.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx328.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Rational</span>
</div>
<div id="lstnumberx329" class="ltx_listingline">
</div>
<div id="lstnumberx330" class="ltx_listingline">
<span id="lstnumberx330.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">def</span><span id="lstnumberx330.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx330.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">octagon_side_length</span><span id="lstnumberx330.4" class="ltx_text ltx_font_typewriter">():</span>
</div>
<div id="lstnumberx331" class="ltx_listingline">
<span id="lstnumberx331.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx331.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">square_side_length</span><span id="lstnumberx331.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx331.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx331.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx331.6" class="ltx_text ltx_font_typewriter">10</span>
</div>
<div id="lstnumberx332" class="ltx_listingline">
<span id="lstnumberx332.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx332.2" class="ltx_text ltx_font_typewriter" style="background-color:#F47874;">octagon_side_length = square_side_length + 5*sqrt(2)/(1 + sqrt(2))</span>
</div>
<div id="lstnumberx333" class="ltx_listingline">
</div>
<div id="lstnumberx334" class="ltx_listingline">
<span id="lstnumberx334.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx334.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx334.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx334.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">round</span><span id="lstnumberx334.5" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx334.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">octagon_side_length</span><span id="lstnumberx334.7" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx334.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx334.9" class="ltx_text ltx_font_typewriter">2)</span>
</div>
<div id="lstnumberx335" class="ltx_listingline">
</div>
<div id="lstnumberx336" class="ltx_listingline">
<span id="lstnumberx336.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx336.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx336.3" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx336.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx336.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">octagon_side_length</span><span id="lstnumberx336.6" class="ltx_text ltx_font_typewriter">()</span>
</div>
<div id="lstnumberx337" class="ltx_listingline">
<span id="lstnumberx337.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">print</span><span id="lstnumberx337.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx337.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx337.4" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx338" class="ltx_listingline">
<span id="lstnumberx338.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx339" class="ltx_listingline">
<span id="lstnumberx339.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx339.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span>
</div>
<div id="lstnumberx340" class="ltx_listingline">
<span id="lstnumberx340.1" class="ltx_text ltx_font_typewriter">12.93</span>
</div>
<div id="lstnumberx341" class="ltx_listingline">
<span id="lstnumberx341.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx342" class="ltx_listingline">
<span id="lstnumberx342.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx342.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx342.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">length</span><span id="lstnumberx342.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx342.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx342.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx342.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">one</span><span id="lstnumberx342.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx342.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">side</span><span id="lstnumberx342.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx342.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx342.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx342.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx342.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx342.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">octagon</span><span id="lstnumberx342.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx342.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx342.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx342.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx342.20" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx342.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx342.22" class="ltx_text ltx_font_typewriter">{12.93}</span><span id="lstnumberx342.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx342.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx342.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">inches</span><span id="lstnumberx342.26" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx343" class="ltx_listingline">
</div>
<div id="lstnumberx344" class="ltx_listingline">
<span id="lstnumberx344.1" class="ltx_text ltx_font_typewriter">**</span><span id="lstnumberx344.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Ground</span><span id="lstnumberx344.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx344.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Truth</span><span id="lstnumberx344.5" class="ltx_text ltx_font_typewriter">**:</span>
</div>
<div id="lstnumberx345" class="ltx_listingline">
<span id="lstnumberx345.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Recall</span><span id="lstnumberx345.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx345.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx345.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ratio</span><span id="lstnumberx345.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx345.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx345.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">hypotenuse</span><span id="lstnumberx345.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx345.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx345.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">leg</span><span id="lstnumberx345.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx345.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">an</span><span id="lstnumberx345.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">isosceles</span><span id="lstnumberx345.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">right</span><span id="lstnumberx345.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">triangle</span><span id="lstnumberx345.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx345.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx345.34" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx345.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx345.36" class="ltx_text ltx_font_typewriter">{2}</span><span id="lstnumberx345.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx345.38" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx345.39" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx345.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Looking</span><span id="lstnumberx345.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">at</span><span id="lstnumberx345.43" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx345.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">removed</span><span id="lstnumberx345.47" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.48" class="ltx_text ltx_lst_identifier ltx_font_typewriter">triangles</span><span id="lstnumberx345.49" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx345.50" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx345.52" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.53" class="ltx_text ltx_lst_identifier ltx_font_typewriter">see</span><span id="lstnumberx345.54" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx345.56" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx345.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">three</span><span id="lstnumberx345.60" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">segments</span><span id="lstnumberx345.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx345.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.65" class="ltx_text ltx_lst_identifier ltx_font_typewriter">make</span><span id="lstnumberx345.66" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">up</span><span id="lstnumberx345.68" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.69" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx345.70" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.71" class="ltx_text ltx_lst_identifier ltx_font_typewriter">side</span><span id="lstnumberx345.72" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.73" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx345.74" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.75" class="ltx_text ltx_lst_identifier ltx_font_typewriter">square</span><span id="lstnumberx345.76" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.77" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx345.78" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.79" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$s</span><span id="lstnumberx345.80" class="ltx_text ltx_font_typewriter">/\</span><span id="lstnumberx345.81" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx345.82" class="ltx_text ltx_font_typewriter">{2}</span><span id="lstnumberx345.83" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx345.84" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx345.85" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.86" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$s$</span><span id="lstnumberx345.87" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx345.88" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.89" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx345.90" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.91" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$s</span><span id="lstnumberx345.92" class="ltx_text ltx_font_typewriter">/\</span><span id="lstnumberx345.93" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx345.94" class="ltx_text ltx_font_typewriter">{2}</span><span id="lstnumberx345.95" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx345.96" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx345.97" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.98" class="ltx_text ltx_lst_identifier ltx_font_typewriter">where</span><span id="lstnumberx345.99" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.100" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$s$</span><span id="lstnumberx345.101" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.102" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx345.103" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.104" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx345.105" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.106" class="ltx_text ltx_lst_identifier ltx_font_typewriter">side</span><span id="lstnumberx345.107" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.108" class="ltx_text ltx_lst_identifier ltx_font_typewriter">length</span><span id="lstnumberx345.109" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.110" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx345.111" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.112" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx345.113" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.114" class="ltx_text ltx_lst_identifier ltx_font_typewriter">octagon</span><span id="lstnumberx345.115" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx345.116" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx345.117" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Setting</span><span id="lstnumberx345.118" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.119" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx345.120" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.121" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx345.122" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.123" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx345.124" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.125" class="ltx_text ltx_lst_identifier ltx_font_typewriter">these</span><span id="lstnumberx345.126" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.127" class="ltx_text ltx_lst_identifier ltx_font_typewriter">three</span><span id="lstnumberx345.128" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.129" class="ltx_text ltx_lst_identifier ltx_font_typewriter">side</span><span id="lstnumberx345.130" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.131" class="ltx_text ltx_lst_identifier ltx_font_typewriter">lengths</span><span id="lstnumberx345.132" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.133" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equal</span><span id="lstnumberx345.134" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.135" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx345.136" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.137" class="ltx_text ltx_font_typewriter">10</span><span id="lstnumberx345.138" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.139" class="ltx_text ltx_lst_identifier ltx_font_typewriter">inches</span><span id="lstnumberx345.140" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx345.141" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.142" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx345.143" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.144" class="ltx_text ltx_lst_identifier ltx_font_typewriter">find</span><span id="lstnumberx345.145" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx345.146" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx345.147" class="ltx_text ltx_lst_identifier ltx_font_typewriter">begin</span><span id="lstnumberx345.148" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx345.149" class="ltx_text ltx_lst_identifier ltx_font_typewriter">align</span><span id="lstnumberx345.150" class="ltx_text ltx_font_typewriter">*}</span>
</div>
<div id="lstnumberx346" class="ltx_listingline">
<span id="lstnumberx346.1" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx346.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx346.3" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx346.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx346.5" class="ltx_text ltx_font_typewriter">}{\</span><span id="lstnumberx346.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx346.7" class="ltx_text ltx_font_typewriter">{2}}+</span><span id="lstnumberx346.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx346.9" class="ltx_text ltx_font_typewriter">+\</span><span id="lstnumberx346.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx346.11" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx346.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx346.13" class="ltx_text ltx_font_typewriter">}{\</span><span id="lstnumberx346.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx346.15" class="ltx_text ltx_font_typewriter">{2}}&amp;=10</span><span id="lstnumberx346.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx346.17" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx346.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">implies</span><span id="lstnumberx346.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx346.20" class="ltx_text ltx_font_typewriter">\\</span>
</div>
<div id="lstnumberx347" class="ltx_listingline">
<span id="lstnumberx347.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx347.2" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx347.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">left</span><span id="lstnumberx347.4" class="ltx_text ltx_font_typewriter">(\</span><span id="lstnumberx347.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx347.6" class="ltx_text ltx_font_typewriter">{1}{\</span><span id="lstnumberx347.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx347.8" class="ltx_text ltx_font_typewriter">{2}}+1+\</span><span id="lstnumberx347.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx347.10" class="ltx_text ltx_font_typewriter">{1}{\</span><span id="lstnumberx347.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx347.12" class="ltx_text ltx_font_typewriter">{2}}\</span><span id="lstnumberx347.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">right</span><span id="lstnumberx347.14" class="ltx_text ltx_font_typewriter">)&amp;=10</span><span id="lstnumberx347.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx347.16" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx347.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">implies</span><span id="lstnumberx347.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx347.19" class="ltx_text ltx_font_typewriter">\\</span>
</div>
<div id="lstnumberx348" class="ltx_listingline">
<span id="lstnumberx348.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx348.2" class="ltx_text ltx_font_typewriter">(\</span><span id="lstnumberx348.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx348.4" class="ltx_text ltx_font_typewriter">{2}+1)&amp;=10</span><span id="lstnumberx348.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx348.6" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx348.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">implies</span><span id="lstnumberx348.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx348.9" class="ltx_text ltx_font_typewriter">\\</span>
</div>
<div id="lstnumberx349" class="ltx_listingline">
<span id="lstnumberx349.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx349.2" class="ltx_text ltx_font_typewriter">&amp;=\</span><span id="lstnumberx349.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx349.4" class="ltx_text ltx_font_typewriter">{10}{\</span><span id="lstnumberx349.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx349.6" class="ltx_text ltx_font_typewriter">{2}+1}=10(\</span><span id="lstnumberx349.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx349.8" class="ltx_text ltx_font_typewriter">{2}-1),</span>
</div>
<div id="lstnumberx350" class="ltx_listingline">
<span id="lstnumberx350.1" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx350.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">end</span><span id="lstnumberx350.3" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx350.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">align</span><span id="lstnumberx350.5" class="ltx_text ltx_font_typewriter">*}</span><span id="lstnumberx350.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx350.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">where</span><span id="lstnumberx350.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx350.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx350.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx350.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">have</span><span id="lstnumberx350.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx350.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rationalized</span><span id="lstnumberx350.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx350.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx350.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx350.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">denominator</span><span id="lstnumberx350.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx350.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">twice</span><span id="lstnumberx350.20" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx350.21" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx350.22" class="ltx_text ltx_font_typewriter">\[</span>
</div>
<div id="lstnumberx351" class="ltx_listingline">
<span id="lstnumberx351.1" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx351.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx351.3" class="ltx_text ltx_font_typewriter">{1}{\</span><span id="lstnumberx351.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx351.5" class="ltx_text ltx_font_typewriter">{2}}=\</span><span id="lstnumberx351.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx351.7" class="ltx_text ltx_font_typewriter">{1}{\</span><span id="lstnumberx351.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx351.9" class="ltx_text ltx_font_typewriter">{2}}\</span><span id="lstnumberx351.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cdot</span><span id="lstnumberx351.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx351.12" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx351.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx351.14" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx351.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx351.16" class="ltx_text ltx_font_typewriter">{2}}{\</span><span id="lstnumberx351.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx351.18" class="ltx_text ltx_font_typewriter">{2}}=\</span><span id="lstnumberx351.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx351.20" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx351.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx351.22" class="ltx_text ltx_font_typewriter">{2}}{2},</span>
</div>
<div id="lstnumberx352" class="ltx_listingline">
<span id="lstnumberx352.1" class="ltx_text ltx_font_typewriter">\]</span><span id="lstnumberx352.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx352.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx352.4" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx352.5" class="ltx_text ltx_font_typewriter">\[</span>
</div>
<div id="lstnumberx353" class="ltx_listingline">
<span id="lstnumberx353.1" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx353.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx353.3" class="ltx_text ltx_font_typewriter">{10}{\</span><span id="lstnumberx353.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx353.5" class="ltx_text ltx_font_typewriter">{2}+1}=\</span><span id="lstnumberx353.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx353.7" class="ltx_text ltx_font_typewriter">{10}{\</span><span id="lstnumberx353.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx353.9" class="ltx_text ltx_font_typewriter">{2}+1}\</span><span id="lstnumberx353.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cdot</span><span id="lstnumberx353.11" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx353.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx353.13" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx353.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx353.15" class="ltx_text ltx_font_typewriter">{2}-1}{\</span><span id="lstnumberx353.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx353.17" class="ltx_text ltx_font_typewriter">{2}-1}=10(\</span><span id="lstnumberx353.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx353.19" class="ltx_text ltx_font_typewriter">{2}-1).</span>
</div>
<div id="lstnumberx354" class="ltx_listingline">
<span id="lstnumberx354.1" class="ltx_text ltx_font_typewriter">\]</span><span id="lstnumberx354.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx354.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">To</span><span id="lstnumberx354.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx354.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx354.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx354.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nearest</span><span id="lstnumberx354.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx354.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">hundredth</span><span id="lstnumberx354.10" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx354.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx354.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$s</span><span id="lstnumberx354.13" class="ltx_text ltx_font_typewriter">=\</span><span id="lstnumberx354.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx354.15" class="ltx_text ltx_font_typewriter">{4.14}</span><span id="lstnumberx354.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx354.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx354.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">inches</span><span id="lstnumberx354.19" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx355" class="ltx_listingline">
</div>
<div id="lstnumberx356" class="ltx_listingline">
<span id="lstnumberx356.1" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx356.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asy</span><span id="lstnumberx356.3" class="ltx_text ltx_font_typewriter">]</span>
</div>
<div id="lstnumberx357" class="ltx_listingline">
<span id="lstnumberx357.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">size</span><span id="lstnumberx357.2" class="ltx_text ltx_font_typewriter">(150);</span>
</div>
<div id="lstnumberx358" class="ltx_listingline">
<span id="lstnumberx358.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">defaultpen</span><span id="lstnumberx358.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx358.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linewidth</span><span id="lstnumberx358.4" class="ltx_text ltx_font_typewriter">(0.7)+</span><span id="lstnumberx358.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">fontsize</span><span id="lstnumberx358.6" class="ltx_text ltx_font_typewriter">(10));</span>
</div>
<div id="lstnumberx359" class="ltx_listingline">
<span id="lstnumberx359.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx359.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx359.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx359.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx359.5" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx359.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx359.7" class="ltx_text ltx_font_typewriter">10/(1+</span><span id="lstnumberx359.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx359.9" class="ltx_text ltx_font_typewriter">(2));</span>
</div>
<div id="lstnumberx360" class="ltx_listingline">
<span id="lstnumberx360.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx360.2" class="ltx_text ltx_font_typewriter">((0,0)--(10,0)--(10,10)--(0,10)--(0,0));</span>
</div>
<div id="lstnumberx361" class="ltx_listingline">
<span id="lstnumberx361.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx361.2" class="ltx_text ltx_font_typewriter">((5*</span><span id="lstnumberx361.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx361.4" class="ltx_text ltx_font_typewriter">(2)/(1</span><span id="lstnumberx361.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx361.6" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx361.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx361.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx361.9" class="ltx_text ltx_font_typewriter">(2)),0)--(0,5*</span><span id="lstnumberx361.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx361.11" class="ltx_text ltx_font_typewriter">(2)/(1</span><span id="lstnumberx361.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx361.13" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx361.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx361.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx361.16" class="ltx_text ltx_font_typewriter">(2))));</span>
</div>
<div id="lstnumberx362" class="ltx_listingline">
<span id="lstnumberx362.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx362.2" class="ltx_text ltx_font_typewriter">((10</span><span id="lstnumberx362.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx362.4" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx362.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx362.6" class="ltx_text ltx_font_typewriter">(5*</span><span id="lstnumberx362.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx362.8" class="ltx_text ltx_font_typewriter">(2)/(1</span><span id="lstnumberx362.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx362.10" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx362.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx362.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx362.13" class="ltx_text ltx_font_typewriter">(2))),0)--(10,5*</span><span id="lstnumberx362.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx362.15" class="ltx_text ltx_font_typewriter">(2)/(1</span><span id="lstnumberx362.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx362.17" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx362.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx362.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx362.20" class="ltx_text ltx_font_typewriter">(2))));</span>
</div>
<div id="lstnumberx363" class="ltx_listingline">
<span id="lstnumberx363.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx363.2" class="ltx_text ltx_font_typewriter">((0,(10-(5*</span><span id="lstnumberx363.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx363.4" class="ltx_text ltx_font_typewriter">(2)/(1+</span><span id="lstnumberx363.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx363.6" class="ltx_text ltx_font_typewriter">(2)))))--(5*</span><span id="lstnumberx363.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx363.8" class="ltx_text ltx_font_typewriter">(2)/(1</span><span id="lstnumberx363.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx363.10" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx363.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx363.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx363.13" class="ltx_text ltx_font_typewriter">(2)),10));</span>
</div>
<div id="lstnumberx364" class="ltx_listingline">
<span id="lstnumberx364.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx364.2" class="ltx_text ltx_font_typewriter">((10</span><span id="lstnumberx364.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx364.4" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx364.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx364.6" class="ltx_text ltx_font_typewriter">(5*</span><span id="lstnumberx364.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx364.8" class="ltx_text ltx_font_typewriter">(2)/(1</span><span id="lstnumberx364.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx364.10" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx364.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx364.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx364.13" class="ltx_text ltx_font_typewriter">(2))),10)--(10,10</span><span id="lstnumberx364.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx364.15" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx364.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx364.17" class="ltx_text ltx_font_typewriter">5*</span><span id="lstnumberx364.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx364.19" class="ltx_text ltx_font_typewriter">(2)/(1</span><span id="lstnumberx364.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx364.21" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx364.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx364.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx364.24" class="ltx_text ltx_font_typewriter">(2))));</span>
</div>
<div id="lstnumberx365" class="ltx_listingline">
</div>
<div id="lstnumberx366" class="ltx_listingline">
<span id="lstnumberx366.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">label</span><span id="lstnumberx366.2" class="ltx_text ltx_font_typewriter">("</span><span id="lstnumberx366.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$s$</span><span id="lstnumberx366.4" class="ltx_text ltx_font_typewriter">",(10-</span><span id="lstnumberx366.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx366.6" class="ltx_text ltx_font_typewriter">/(2*</span><span id="lstnumberx366.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx366.8" class="ltx_text ltx_font_typewriter">(2)),10-</span><span id="lstnumberx366.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx366.10" class="ltx_text ltx_font_typewriter">/(2*</span><span id="lstnumberx366.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx366.12" class="ltx_text ltx_font_typewriter">(2))),</span><span id="lstnumberx366.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">SW</span><span id="lstnumberx366.14" class="ltx_text ltx_font_typewriter">);</span>
</div>
<div id="lstnumberx367" class="ltx_listingline">
<span id="lstnumberx367.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">label</span><span id="lstnumberx367.2" class="ltx_text ltx_font_typewriter">("</span><span id="lstnumberx367.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx367.4" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx367.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">displaystyle</span><span id="lstnumberx367.6" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx367.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx367.8" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx367.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx367.10" class="ltx_text ltx_font_typewriter">}{\</span><span id="lstnumberx367.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx367.12" class="ltx_text ltx_font_typewriter">{2}}}</span><span id="lstnumberx367.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx367.14" class="ltx_text ltx_font_typewriter">",(10,10-</span><span id="lstnumberx367.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx367.16" class="ltx_text ltx_font_typewriter">/(2*</span><span id="lstnumberx367.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx367.18" class="ltx_text ltx_font_typewriter">(2))),</span><span id="lstnumberx367.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">E</span><span id="lstnumberx367.20" class="ltx_text ltx_font_typewriter">);</span>
</div>
<div id="lstnumberx368" class="ltx_listingline">
<span id="lstnumberx368.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">label</span><span id="lstnumberx368.2" class="ltx_text ltx_font_typewriter">("</span><span id="lstnumberx368.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx368.4" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx368.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">displaystyle</span><span id="lstnumberx368.6" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx368.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx368.8" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx368.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx368.10" class="ltx_text ltx_font_typewriter">}{\</span><span id="lstnumberx368.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx368.12" class="ltx_text ltx_font_typewriter">{2}}}</span><span id="lstnumberx368.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx368.14" class="ltx_text ltx_font_typewriter">",(10,</span><span id="lstnumberx368.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx368.16" class="ltx_text ltx_font_typewriter">/(2*</span><span id="lstnumberx368.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx368.18" class="ltx_text ltx_font_typewriter">(2))),</span><span id="lstnumberx368.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">E</span><span id="lstnumberx368.20" class="ltx_text ltx_font_typewriter">);</span>
</div>
<div id="lstnumberx369" class="ltx_listingline">
<span id="lstnumberx369.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">label</span><span id="lstnumberx369.2" class="ltx_text ltx_font_typewriter">("</span><span id="lstnumberx369.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$s$</span><span id="lstnumberx369.4" class="ltx_text ltx_font_typewriter">",(10,5),</span><span id="lstnumberx369.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">E</span><span id="lstnumberx369.6" class="ltx_text ltx_font_typewriter">);</span>
</div>
<div id="lstnumberx370" class="ltx_listingline">
</div>
<div id="lstnumberx371" class="ltx_listingline">
<span id="lstnumberx371.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx371.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx371.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rightanglemark</span><span id="lstnumberx371.4" class="ltx_text ltx_font_typewriter">((10,0),(10,10),(0,10)));[/</span><span id="lstnumberx371.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asy</span><span id="lstnumberx371.6" class="ltx_text ltx_font_typewriter">]</span>
</div>
</div>
</figure>
<figure id="LST8" class="ltx_float ltx_lstlisting">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float">Listing 8: </span>Failure case for <span id="LST8.3.1" class="ltx_text ltx_font_smallcaps">ToRA</span>: <span id="LST8.4.2" class="ltx_text ltx_font_bold">Inappropriate Tool Use</span>.</figcaption>
<div id="LST8.5" class="ltx_listing ltx_lstlisting ltx_framed ltx_framed_topbottom ltx_listing">
<div class="ltx_listing_data"><a href="data:text/plain;base64,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" download="section/cases/failure/4_inappropriate_tool_use.md">⬇</a></div>
<div id="lstnumberx372" class="ltx_listingline">
<span id="lstnumberx372.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Idx</span><span id="lstnumberx372.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx372.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx372.4" class="ltx_text ltx_font_typewriter">2489</span>
</div>
<div id="lstnumberx373" class="ltx_listingline">
<span id="lstnumberx373.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Type</span><span id="lstnumberx373.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx373.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx373.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Intermediate</span><span id="lstnumberx373.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx373.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Algebra</span>
</div>
<div id="lstnumberx374" class="ltx_listingline">
<span id="lstnumberx374.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Level</span><span id="lstnumberx374.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx374.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx374.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Level</span><span id="lstnumberx374.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx374.6" class="ltx_text ltx_font_typewriter">5</span>
</div>
<div id="lstnumberx375" class="ltx_listingline">
<span id="lstnumberx375.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Question</span><span id="lstnumberx375.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx375.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.4" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx375.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Let</span><span id="lstnumberx375.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_1</span><span id="lstnumberx375.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx375.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx375.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_2</span><span id="lstnumberx375.12" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx375.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx375.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_3$</span><span id="lstnumberx375.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">be</span><span id="lstnumberx375.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">complex</span><span id="lstnumberx375.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">numbers</span><span id="lstnumberx375.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">such</span><span id="lstnumberx375.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx375.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx375.28" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx375.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx375.30" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx375.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.32" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx375.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.34" class="ltx_text ltx_font_typewriter">2,</span><span id="lstnumberx375.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx375.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx375.38" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx375.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx375.40" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx375.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.42" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx375.43" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.44" class="ltx_text ltx_font_typewriter">3,</span><span id="lstnumberx375.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx375.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx375.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx375.50" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx375.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx375.52" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx375.53" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.54" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx375.55" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.56" class="ltx_text ltx_font_typewriter">4.</span><span id="lstnumberx375.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx375.58" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx375.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Find</span><span id="lstnumberx375.60" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx375.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">largest</span><span id="lstnumberx375.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.65" class="ltx_text ltx_lst_identifier ltx_font_typewriter">possible</span><span id="lstnumberx375.66" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">value</span><span id="lstnumberx375.68" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.69" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx375.70" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx375.71" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx375.72" class="ltx_text ltx_font_typewriter">\\[|</span><span id="lstnumberx375.73" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx375.74" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.75" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx375.76" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.77" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx375.78" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx375.79" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.80" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx375.81" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.82" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx375.83" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx375.84" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.85" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx375.86" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.87" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx375.88" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx375.89" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.90" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx375.91" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.92" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx375.93" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx375.94" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.95" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx375.96" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx375.97" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx375.98" class="ltx_text ltx_font_typewriter">|^2.\\]’</span>
</div>
<div id="lstnumberx376" class="ltx_listingline">
<span id="lstnumberx376.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solution</span><span id="lstnumberx376.2" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx377" class="ltx_listingline">
<span id="lstnumberx377.1" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx377.2" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx377.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span>
</div>
<div id="lstnumberx378" class="ltx_listingline">
<span id="lstnumberx378.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx378.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx378.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx378.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx378.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx378.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx378.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span><span id="lstnumberx378.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx378.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx378.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">simplify</span><span id="lstnumberx378.11" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx378.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx378.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Max</span>
</div>
<div id="lstnumberx379" class="ltx_listingline">
<span id="lstnumberx379.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx379.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx379.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx379.4" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx379.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">functions</span><span id="lstnumberx379.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx379.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx379.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx379.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Abs</span>
</div>
<div id="lstnumberx380" class="ltx_listingline">
</div>
<div id="lstnumberx381" class="ltx_listingline">
<span id="lstnumberx381.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">def</span><span id="lstnumberx381.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx381.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">max_expression_value</span><span id="lstnumberx381.4" class="ltx_text ltx_font_typewriter">():</span>
</div>
<div id="lstnumberx382" class="ltx_listingline">
<span id="lstnumberx382.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx382.2" class="ltx_text ltx_font_typewriter">"""</span><span id="lstnumberx382.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Let</span><span id="lstnumberx382.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_1</span><span id="lstnumberx382.6" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx382.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx382.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_2</span><span id="lstnumberx382.10" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx382.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx382.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_3$</span><span id="lstnumberx382.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">be</span><span id="lstnumberx382.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">complex</span><span id="lstnumberx382.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">numbers</span><span id="lstnumberx382.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">such</span><span id="lstnumberx382.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx382.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx382.26" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx382.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx382.28" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx382.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.30" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx382.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.32" class="ltx_text ltx_font_typewriter">2,</span><span id="lstnumberx382.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx382.34" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx382.36" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx382.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx382.38" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx382.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.40" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx382.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.42" class="ltx_text ltx_font_typewriter">3,</span><span id="lstnumberx382.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx382.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx382.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx382.48" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx382.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx382.50" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx382.51" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.52" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx382.53" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.54" class="ltx_text ltx_font_typewriter">4.</span><span id="lstnumberx382.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx382.56" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx382.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Find</span><span id="lstnumberx382.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx382.60" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">largest</span><span id="lstnumberx382.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">possible</span><span id="lstnumberx382.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.65" class="ltx_text ltx_lst_identifier ltx_font_typewriter">value</span><span id="lstnumberx382.66" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx382.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span>
</div>
<div id="lstnumberx383" class="ltx_listingline">
<span id="lstnumberx383.1" class="ltx_text ltx_font_typewriter">\[|</span><span id="lstnumberx383.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx383.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx383.4" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx383.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx383.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx383.7" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx383.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx383.9" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx383.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx383.11" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx383.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx383.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx383.14" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx383.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx383.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx383.17" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx383.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx383.19" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx383.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx383.21" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx383.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx383.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx383.24" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx383.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx383.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx383.27" class="ltx_text ltx_font_typewriter">|^2.\]"""</span>
</div>
<div id="lstnumberx384" class="ltx_listingline">
<span id="lstnumberx384.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx384.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx384.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx384.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Define</span><span id="lstnumberx384.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx384.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx384.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx384.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">variables</span>
</div>
<div id="lstnumberx385" class="ltx_listingline">
<span id="lstnumberx385.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx385.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z1</span><span id="lstnumberx385.3" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx385.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx385.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z2</span><span id="lstnumberx385.6" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx385.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx385.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z3</span><span id="lstnumberx385.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx385.10" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx385.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx385.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span><span id="lstnumberx385.13" class="ltx_text ltx_font_typewriter">(’</span><span id="lstnumberx385.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z1</span><span id="lstnumberx385.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx385.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z2</span><span id="lstnumberx385.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx385.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z3</span><span id="lstnumberx385.19" class="ltx_text ltx_font_typewriter">’)</span>
</div>
<div id="lstnumberx386" class="ltx_listingline">
</div>
<div id="lstnumberx387" class="ltx_listingline">
<span id="lstnumberx387.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx387.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx387.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx387.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Define</span><span id="lstnumberx387.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx387.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx387.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx387.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">expression</span>
</div>
<div id="lstnumberx388" class="ltx_listingline">
<span id="lstnumberx388.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx388.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">expr</span><span id="lstnumberx388.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx388.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx388.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx388.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Abs</span><span id="lstnumberx388.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx388.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z1</span><span id="lstnumberx388.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx388.10" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx388.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx388.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z2</span><span id="lstnumberx388.13" class="ltx_text ltx_font_typewriter">)**2</span><span id="lstnumberx388.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx388.15" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx388.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx388.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Abs</span><span id="lstnumberx388.18" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx388.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z1</span><span id="lstnumberx388.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx388.21" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx388.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx388.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z3</span><span id="lstnumberx388.24" class="ltx_text ltx_font_typewriter">)**2</span><span id="lstnumberx388.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx388.26" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx388.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx388.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Abs</span><span id="lstnumberx388.29" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx388.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z2</span><span id="lstnumberx388.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx388.32" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx388.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx388.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z3</span><span id="lstnumberx388.35" class="ltx_text ltx_font_typewriter">)**2</span>
</div>
<div id="lstnumberx389" class="ltx_listingline">
</div>
<div id="lstnumberx390" class="ltx_listingline">
<span id="lstnumberx390.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx390.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx390.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx390.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Define</span><span id="lstnumberx390.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx390.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx390.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx390.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">conditions</span>
</div>
<div id="lstnumberx391" class="ltx_listingline">
<span id="lstnumberx391.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx391.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">conditions</span><span id="lstnumberx391.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx391.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx391.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx391.6" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx391.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Abs</span><span id="lstnumberx391.8" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx391.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z1</span><span id="lstnumberx391.10" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx391.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx391.12" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx391.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx391.14" class="ltx_text ltx_font_typewriter">2,</span><span id="lstnumberx391.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx391.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Abs</span><span id="lstnumberx391.17" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx391.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z2</span><span id="lstnumberx391.19" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx391.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx391.21" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx391.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx391.23" class="ltx_text ltx_font_typewriter">3,</span><span id="lstnumberx391.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx391.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Abs</span><span id="lstnumberx391.26" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx391.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z3</span><span id="lstnumberx391.28" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx391.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx391.30" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx391.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx391.32" class="ltx_text ltx_font_typewriter">4]</span>
</div>
<div id="lstnumberx392" class="ltx_listingline">
</div>
<div id="lstnumberx393" class="ltx_listingline">
<span id="lstnumberx393.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx393.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx393.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx393.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Find</span><span id="lstnumberx393.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx393.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx393.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx393.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">maximum</span><span id="lstnumberx393.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx393.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">value</span><span id="lstnumberx393.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx393.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx393.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx393.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx393.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx393.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">expression</span><span id="lstnumberx393.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx393.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">under</span><span id="lstnumberx393.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx393.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx393.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx393.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">conditions</span>
</div>
<div id="lstnumberx394" class="ltx_listingline">
<span id="lstnumberx394.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx394.2" class="ltx_text ltx_font_typewriter" style="background-color:#F47874;">max_value = Max(expr, *conditions)</span>
</div>
<div id="lstnumberx395" class="ltx_listingline">
</div>
<div id="lstnumberx396" class="ltx_listingline">
<span id="lstnumberx396.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx396.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx396.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx396.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Simplify</span><span id="lstnumberx396.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx396.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx396.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx396.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">maximum</span><span id="lstnumberx396.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx396.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">value</span>
</div>
<div id="lstnumberx397" class="ltx_listingline">
<span id="lstnumberx397.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx397.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">simplified_max_value</span><span id="lstnumberx397.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx397.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx397.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx397.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">simplify</span><span id="lstnumberx397.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx397.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">max_value</span><span id="lstnumberx397.9" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx398" class="ltx_listingline">
</div>
<div id="lstnumberx399" class="ltx_listingline">
<span id="lstnumberx399.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx399.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx399.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx399.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">simplified_max_value</span>
</div>
<div id="lstnumberx400" class="ltx_listingline">
</div>
<div id="lstnumberx401" class="ltx_listingline">
<span id="lstnumberx401.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx401.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx401.3" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx401.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx401.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">max_expression_value</span><span id="lstnumberx401.6" class="ltx_text ltx_font_typewriter">()</span>
</div>
<div id="lstnumberx402" class="ltx_listingline">
<span id="lstnumberx402.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">print</span><span id="lstnumberx402.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx402.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx402.4" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx403" class="ltx_listingline">
<span id="lstnumberx403.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx404" class="ltx_listingline">
<span id="lstnumberx404.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx404.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span>
</div>
<div id="lstnumberx405" class="ltx_listingline">
<span id="lstnumberx405.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Max</span><span id="lstnumberx405.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx405.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Abs</span><span id="lstnumberx405.4" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx405.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z1</span><span id="lstnumberx405.6" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx405.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.8" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx405.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.10" class="ltx_text ltx_font_typewriter">2,</span><span id="lstnumberx405.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Abs</span><span id="lstnumberx405.13" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx405.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z2</span><span id="lstnumberx405.15" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx405.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.17" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx405.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.19" class="ltx_text ltx_font_typewriter">3,</span><span id="lstnumberx405.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Abs</span><span id="lstnumberx405.22" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx405.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z3</span><span id="lstnumberx405.24" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx405.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.26" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx405.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.28" class="ltx_text ltx_font_typewriter">4,</span><span id="lstnumberx405.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Abs</span><span id="lstnumberx405.31" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx405.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z1</span><span id="lstnumberx405.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.34" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx405.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z2</span><span id="lstnumberx405.37" class="ltx_text ltx_font_typewriter">)**2</span><span id="lstnumberx405.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.39" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx405.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Abs</span><span id="lstnumberx405.42" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx405.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z1</span><span id="lstnumberx405.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.45" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx405.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z3</span><span id="lstnumberx405.48" class="ltx_text ltx_font_typewriter">)**2</span><span id="lstnumberx405.49" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.50" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx405.51" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Abs</span><span id="lstnumberx405.53" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx405.54" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z2</span><span id="lstnumberx405.55" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.56" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx405.57" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx405.58" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z3</span><span id="lstnumberx405.59" class="ltx_text ltx_font_typewriter">)**2)</span>
</div>
<div id="lstnumberx406" class="ltx_listingline">
<span id="lstnumberx406.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx407" class="ltx_listingline">
<span id="lstnumberx407.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx407.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx407.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">largest</span><span id="lstnumberx407.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx407.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">possible</span><span id="lstnumberx407.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx407.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">value</span><span id="lstnumberx407.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx407.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx407.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx407.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx407.12" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx407.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx407.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx407.15" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx407.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx407.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx407.18" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx407.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx407.20" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx407.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx407.22" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx407.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx407.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx407.25" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx407.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx407.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx407.28" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx407.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx407.30" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx407.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx407.32" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx407.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx407.34" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx407.35" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx407.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx407.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx407.38" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx407.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx407.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx407.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx407.42" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx407.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx407.44" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx407.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx407.46" class="ltx_text ltx_font_typewriter">{25}</span><span id="lstnumberx407.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx407.48" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx408" class="ltx_listingline">
</div>
<div id="lstnumberx409" class="ltx_listingline">
<span id="lstnumberx409.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Execution</span><span id="lstnumberx409.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx409.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx409.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">None</span>
</div>
<div id="lstnumberx410" class="ltx_listingline">
<span id="lstnumberx410.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Prediction</span><span id="lstnumberx410.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx410.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx410.4" class="ltx_text ltx_font_typewriter">’25’</span>
</div>
<div id="lstnumberx411" class="ltx_listingline">
<span id="lstnumberx411.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Ground</span><span id="lstnumberx411.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx411.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Truth</span><span id="lstnumberx411.4" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx411.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx411.6" class="ltx_text ltx_font_typewriter">’87’</span>
</div>
<div id="lstnumberx412" class="ltx_listingline">
<span id="lstnumberx412.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Score</span><span id="lstnumberx412.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx412.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx412.4" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx412.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">False</span><span id="lstnumberx412.6" class="ltx_text ltx_font_typewriter">]</span>
</div>
<div id="lstnumberx413" class="ltx_listingline">
<span id="lstnumberx413.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Reference</span><span id="lstnumberx413.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">CoT</span><span id="lstnumberx413.4" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx413.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.6" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx413.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">We</span><span id="lstnumberx413.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">have</span><span id="lstnumberx413.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx413.12" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx413.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx413.14" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">begin</span><span id="lstnumberx413.16" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">align</span><span id="lstnumberx413.18" class="ltx_text ltx_font_typewriter">*}\</span><span id="lstnumberx413.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nz_1</span><span id="lstnumberx413.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.21" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.23" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.25" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_1</span><span id="lstnumberx413.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.28" class="ltx_text ltx_font_typewriter">&amp;=</span><span id="lstnumberx413.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.30" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx413.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx413.32" class="ltx_text ltx_font_typewriter">|^2,</span><span id="lstnumberx413.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.34" class="ltx_text ltx_font_typewriter">\\\\\</span><span id="lstnumberx413.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nz_2</span><span id="lstnumberx413.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.37" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.39" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.41" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_2</span><span id="lstnumberx413.43" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.44" class="ltx_text ltx_font_typewriter">&amp;=</span><span id="lstnumberx413.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.46" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx413.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx413.48" class="ltx_text ltx_font_typewriter">|^2,</span><span id="lstnumberx413.49" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.50" class="ltx_text ltx_font_typewriter">\\\\\</span><span id="lstnumberx413.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nz_3</span><span id="lstnumberx413.52" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.53" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.54" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.55" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.56" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.57" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.58" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_3</span><span id="lstnumberx413.59" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.60" class="ltx_text ltx_font_typewriter">&amp;=</span><span id="lstnumberx413.61" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.62" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx413.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx413.64" class="ltx_text ltx_font_typewriter">|^2.\</span><span id="lstnumberx413.65" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx413.66" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">end</span><span id="lstnumberx413.68" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.69" class="ltx_text ltx_lst_identifier ltx_font_typewriter">align</span><span id="lstnumberx413.70" class="ltx_text ltx_font_typewriter">*}</span><span id="lstnumberx413.71" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Likewise</span><span id="lstnumberx413.72" class="ltx_text ltx_font_typewriter">,\</span><span id="lstnumberx413.73" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx413.74" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.75" class="ltx_text ltx_lst_identifier ltx_font_typewriter">begin</span><span id="lstnumberx413.76" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.77" class="ltx_text ltx_lst_identifier ltx_font_typewriter">align</span><span id="lstnumberx413.78" class="ltx_text ltx_font_typewriter">*}\</span><span id="lstnumberx413.79" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx413.80" class="ltx_text ltx_font_typewriter">&amp;|</span><span id="lstnumberx413.81" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx413.82" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.83" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.84" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.85" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx413.86" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx413.87" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.88" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.89" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.90" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx413.91" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx413.92" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.93" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.94" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.95" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx413.96" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx413.97" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.98" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.99" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.100" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx413.101" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx413.102" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.103" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.104" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.105" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx413.106" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx413.107" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.108" class="ltx_text ltx_font_typewriter">\\\\\</span><span id="lstnumberx413.109" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx413.110" class="ltx_text ltx_font_typewriter">&amp;=</span><span id="lstnumberx413.111" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.112" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx413.113" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx413.114" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.115" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.116" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.117" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx413.118" class="ltx_text ltx_font_typewriter">)(\\</span><span id="lstnumberx413.119" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.120" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.121" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx413.122" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.123" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.124" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.125" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx413.126" class="ltx_text ltx_font_typewriter">})</span><span id="lstnumberx413.127" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.128" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.129" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.130" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx413.131" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx413.132" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.133" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.134" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.135" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx413.136" class="ltx_text ltx_font_typewriter">)(\\</span><span id="lstnumberx413.137" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.138" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.139" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx413.140" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.141" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.142" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.143" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx413.144" class="ltx_text ltx_font_typewriter">})</span><span id="lstnumberx413.145" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.146" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.147" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.148" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx413.149" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx413.150" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.151" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.152" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.153" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx413.154" class="ltx_text ltx_font_typewriter">)(\\</span><span id="lstnumberx413.155" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.156" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.157" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx413.158" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.159" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.160" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.161" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx413.162" class="ltx_text ltx_font_typewriter">})</span><span id="lstnumberx413.163" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.164" class="ltx_text ltx_font_typewriter">\\\\\</span><span id="lstnumberx413.165" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx413.166" class="ltx_text ltx_font_typewriter">&amp;=</span><span id="lstnumberx413.167" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.168" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx413.169" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx413.170" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.171" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.172" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.173" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx413.174" class="ltx_text ltx_font_typewriter">)(\\</span><span id="lstnumberx413.175" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.176" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.177" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.178" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.179" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_1</span><span id="lstnumberx413.180" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.181" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.182" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.183" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.184" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.185" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.186" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.187" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.188" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_2</span><span id="lstnumberx413.189" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx413.190" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.191" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.192" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.193" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx413.194" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx413.195" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.196" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.197" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.198" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx413.199" class="ltx_text ltx_font_typewriter">)(\\</span><span id="lstnumberx413.200" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.201" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.202" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.203" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.204" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_1</span><span id="lstnumberx413.205" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.206" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.207" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.208" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.209" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.210" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.211" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.212" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.213" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_3</span><span id="lstnumberx413.214" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx413.215" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.216" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.217" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.218" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx413.219" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx413.220" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.221" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.222" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.223" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx413.224" class="ltx_text ltx_font_typewriter">)(\\</span><span id="lstnumberx413.225" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.226" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.227" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.228" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.229" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_2</span><span id="lstnumberx413.230" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.231" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.232" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.233" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.234" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.235" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.236" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.237" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.238" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_3</span><span id="lstnumberx413.239" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx413.240" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.241" class="ltx_text ltx_font_typewriter">\\\\\</span><span id="lstnumberx413.242" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx413.243" class="ltx_text ltx_font_typewriter">&amp;=</span><span id="lstnumberx413.244" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.245" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx413.246" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.247" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.248" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.249" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.250" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.251" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.252" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_1</span><span id="lstnumberx413.253" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.254" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.255" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.256" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx413.257" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.258" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.259" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.260" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.261" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.262" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.263" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_2</span><span id="lstnumberx413.264" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.265" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.266" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.267" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.268" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.269" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.270" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.271" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.272" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_1</span><span id="lstnumberx413.273" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.274" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx413.275" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.276" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.277" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.278" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx413.279" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.280" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.281" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.282" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.283" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.284" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.285" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_2</span><span id="lstnumberx413.286" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.287" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.288" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.289" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx413.290" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.291" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.292" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.293" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.294" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.295" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.296" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_1</span><span id="lstnumberx413.297" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.298" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.299" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.300" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx413.301" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.302" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.303" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.304" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.305" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.306" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.307" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_3</span><span id="lstnumberx413.308" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.309" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.310" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.311" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.312" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.313" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.314" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.315" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.316" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_1</span><span id="lstnumberx413.317" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.318" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx413.319" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.320" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.321" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.322" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx413.323" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.324" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.325" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.326" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.327" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.328" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.329" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_3</span><span id="lstnumberx413.330" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.331" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.332" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.333" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx413.334" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.335" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.336" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.337" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.338" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.339" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.340" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_3</span><span id="lstnumberx413.341" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.342" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.343" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.344" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx413.345" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.346" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.347" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.348" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.349" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.350" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.351" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_3</span><span id="lstnumberx413.352" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.353" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.354" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.355" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.356" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.357" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.358" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.359" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.360" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_2</span><span id="lstnumberx413.361" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.362" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx413.363" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.364" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.365" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.366" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx413.367" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.368" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.369" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.370" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.371" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.372" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.373" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_3</span><span id="lstnumberx413.374" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.375" class="ltx_text ltx_font_typewriter">\\\\\</span><span id="lstnumberx413.376" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx413.377" class="ltx_text ltx_font_typewriter">&amp;=</span><span id="lstnumberx413.378" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.379" class="ltx_text ltx_font_typewriter">2|</span><span id="lstnumberx413.380" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx413.381" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx413.382" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.383" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.384" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.385" class="ltx_text ltx_font_typewriter">2|</span><span id="lstnumberx413.386" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx413.387" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx413.388" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.389" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.390" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.391" class="ltx_text ltx_font_typewriter">2|</span><span id="lstnumberx413.392" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx413.393" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx413.394" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.395" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx413.396" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.397" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx413.398" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx413.399" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.400" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.401" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.402" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.403" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.404" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.405" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_2</span><span id="lstnumberx413.406" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.407" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.408" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.409" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.410" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.411" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.412" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.413" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.414" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_1</span><span id="lstnumberx413.415" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.416" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx413.417" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.418" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.419" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.420" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx413.421" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.422" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.423" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.424" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.425" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.426" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.427" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_3</span><span id="lstnumberx413.428" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.429" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.430" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.431" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.432" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.433" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.434" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.435" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.436" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_1</span><span id="lstnumberx413.437" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.438" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx413.439" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.440" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.441" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.442" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx413.443" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.444" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.445" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.446" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.447" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.448" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.449" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_3</span><span id="lstnumberx413.450" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.451" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx413.452" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.453" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.454" class="ltx_text ltx_lst_identifier ltx_font_typewriter">overline</span><span id="lstnumberx413.455" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.456" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z</span><span id="lstnumberx413.457" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx413.458" class="ltx_text ltx_lst_identifier ltx_font_typewriter">_2</span><span id="lstnumberx413.459" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx413.460" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx413.461" class="ltx_text ltx_font_typewriter">).\</span><span id="lstnumberx413.462" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx413.463" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx413.464" class="ltx_text ltx_lst_identifier ltx_font_typewriter">end</span><span id="lstnumberx413.465" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx413.466" class="ltx_text ltx_lst_identifier ltx_font_typewriter">align</span><span id="lstnumberx413.467" class="ltx_text ltx_font_typewriter">*}</span>
</div>
<div id="lstnumberx414" class="ltx_listingline">
<span id="lstnumberx414.1" class="ltx_text ltx_font_typewriter">...</span>
</div>
<div id="lstnumberx415" class="ltx_listingline">
<span id="lstnumberx415.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Adding</span><span id="lstnumberx415.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">these</span><span id="lstnumberx415.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">two</span><span id="lstnumberx415.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equations</span><span id="lstnumberx415.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx415.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx415.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">get</span><span id="lstnumberx415.13" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx415.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx415.15" class="ltx_text ltx_font_typewriter">\\[|</span><span id="lstnumberx415.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx415.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.18" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx415.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx415.21" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx415.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.23" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx415.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.25" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx415.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx415.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.28" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx415.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx415.31" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx415.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.33" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx415.34" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.35" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx415.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx415.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.38" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx415.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx415.41" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx415.42" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.43" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx415.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.45" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx415.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx415.47" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.48" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx415.49" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.50" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx415.51" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.52" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx415.53" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.54" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx415.55" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx415.56" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.57" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx415.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.59" class="ltx_text ltx_font_typewriter">3|</span><span id="lstnumberx415.60" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx415.61" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx415.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.63" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx415.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.65" class="ltx_text ltx_font_typewriter">3|</span><span id="lstnumberx415.66" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx415.67" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx415.68" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.69" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx415.70" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.71" class="ltx_text ltx_font_typewriter">3|</span><span id="lstnumberx415.72" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx415.73" class="ltx_text ltx_font_typewriter">|^2.\\]</span><span id="lstnumberx415.74" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Therefore</span><span id="lstnumberx415.75" class="ltx_text ltx_font_typewriter">,\</span><span id="lstnumberx415.76" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx415.77" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx415.78" class="ltx_text ltx_lst_identifier ltx_font_typewriter">begin</span><span id="lstnumberx415.79" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx415.80" class="ltx_text ltx_lst_identifier ltx_font_typewriter">align</span><span id="lstnumberx415.81" class="ltx_text ltx_font_typewriter">*}\</span><span id="lstnumberx415.82" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx415.83" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx415.84" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx415.85" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.86" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx415.87" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.88" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx415.89" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx415.90" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.91" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx415.92" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.93" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx415.94" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx415.95" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.96" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx415.97" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.98" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx415.99" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx415.100" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.101" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx415.102" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.103" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx415.104" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx415.105" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.106" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx415.107" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.108" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx415.109" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx415.110" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.111" class="ltx_text ltx_font_typewriter">&amp;=</span><span id="lstnumberx415.112" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.113" class="ltx_text ltx_font_typewriter">3|</span><span id="lstnumberx415.114" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx415.115" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx415.116" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.117" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx415.118" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.119" class="ltx_text ltx_font_typewriter">3|</span><span id="lstnumberx415.120" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx415.121" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx415.122" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.123" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx415.124" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.125" class="ltx_text ltx_font_typewriter">3|</span><span id="lstnumberx415.126" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx415.127" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx415.128" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.129" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx415.130" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.131" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx415.132" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_1</span><span id="lstnumberx415.133" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.134" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx415.135" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.136" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx415.137" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.138" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx415.139" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.140" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx415.141" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx415.142" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.143" class="ltx_text ltx_font_typewriter">\\\\\</span><span id="lstnumberx415.144" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx415.145" class="ltx_text ltx_font_typewriter">&amp;\\</span><span id="lstnumberx415.146" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx415.147" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.148" class="ltx_text ltx_font_typewriter">3</span><span id="lstnumberx415.149" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.150" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx415.151" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cdot</span><span id="lstnumberx415.152" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.153" class="ltx_text ltx_font_typewriter">2^2</span><span id="lstnumberx415.154" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.155" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx415.156" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.157" class="ltx_text ltx_font_typewriter">3</span><span id="lstnumberx415.158" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.159" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx415.160" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cdot</span><span id="lstnumberx415.161" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.162" class="ltx_text ltx_font_typewriter">3^2</span><span id="lstnumberx415.163" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.164" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx415.165" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.166" class="ltx_text ltx_font_typewriter">3</span><span id="lstnumberx415.167" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.168" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx415.169" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cdot</span><span id="lstnumberx415.170" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.171" class="ltx_text ltx_font_typewriter">4^2</span><span id="lstnumberx415.172" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.173" class="ltx_text ltx_font_typewriter">\\\\\</span><span id="lstnumberx415.174" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx415.175" class="ltx_text ltx_font_typewriter">&amp;=</span><span id="lstnumberx415.176" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.177" class="ltx_text ltx_font_typewriter">87.\</span><span id="lstnumberx415.178" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx415.179" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx415.180" class="ltx_text ltx_lst_identifier ltx_font_typewriter">end</span><span id="lstnumberx415.181" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx415.182" class="ltx_text ltx_lst_identifier ltx_font_typewriter">align</span><span id="lstnumberx415.183" class="ltx_text ltx_font_typewriter">*}</span><span id="lstnumberx415.184" class="ltx_text ltx_lst_identifier ltx_font_typewriter">For</span><span id="lstnumberx415.185" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.186" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equality</span><span id="lstnumberx415.187" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.188" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx415.189" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.190" class="ltx_text ltx_lst_identifier ltx_font_typewriter">occur</span><span id="lstnumberx415.191" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx415.192" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.193" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx415.194" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.195" class="ltx_text ltx_lst_identifier ltx_font_typewriter">must</span><span id="lstnumberx415.196" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.197" class="ltx_text ltx_lst_identifier ltx_font_typewriter">have</span><span id="lstnumberx415.198" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.199" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_1</span><span id="lstnumberx415.200" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.201" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx415.202" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.203" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx415.204" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.205" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx415.206" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.207" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx415.208" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.209" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx415.210" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.211" class="ltx_text ltx_font_typewriter">0.</span><span id="lstnumberx415.212" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx415.213" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx415.214" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Without</span><span id="lstnumberx415.215" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.216" class="ltx_text ltx_lst_identifier ltx_font_typewriter">loss</span><span id="lstnumberx415.217" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.218" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx415.219" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.220" class="ltx_text ltx_lst_identifier ltx_font_typewriter">generality</span><span id="lstnumberx415.221" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx415.222" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.223" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx415.224" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.225" class="ltx_text ltx_lst_identifier ltx_font_typewriter">can</span><span id="lstnumberx415.226" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.227" class="ltx_text ltx_lst_identifier ltx_font_typewriter">assume</span><span id="lstnumberx415.228" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.229" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx415.230" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.231" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_1</span><span id="lstnumberx415.232" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.233" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx415.234" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx415.235" class="ltx_text ltx_font_typewriter">2.</span><span id="lstnumberx415.236" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span>
</div>
<div id="lstnumberx416" class="ltx_listingline">
<span id="lstnumberx416.1" class="ltx_text ltx_font_typewriter">...</span>
</div>
<div id="lstnumberx417" class="ltx_listingline">
<span id="lstnumberx417.1" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx417.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asy</span><span id="lstnumberx417.3" class="ltx_text ltx_font_typewriter">]\</span><span id="lstnumberx417.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nunitsize</span><span id="lstnumberx417.5" class="ltx_text ltx_font_typewriter">(1</span><span id="lstnumberx417.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cm</span><span id="lstnumberx417.8" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx417.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx417.10" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx417.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">npair</span><span id="lstnumberx417.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">zone</span><span id="lstnumberx417.14" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx417.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ztwo</span><span id="lstnumberx417.17" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx417.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">zthree</span><span id="lstnumberx417.20" class="ltx_text ltx_font_typewriter">;\</span><span id="lstnumberx417.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx417.22" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx417.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nzone</span><span id="lstnumberx417.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.25" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx417.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.27" class="ltx_text ltx_font_typewriter">(2,0);\</span><span id="lstnumberx417.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nztwo</span><span id="lstnumberx417.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.30" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx417.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.32" class="ltx_text ltx_font_typewriter">(3/4,3*</span><span id="lstnumberx417.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx417.34" class="ltx_text ltx_font_typewriter">(15)/4);\</span><span id="lstnumberx417.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nzthree</span><span id="lstnumberx417.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.37" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx417.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.39" class="ltx_text ltx_font_typewriter">(-11/4,-3*</span><span id="lstnumberx417.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx417.41" class="ltx_text ltx_font_typewriter">(15)/4);\</span><span id="lstnumberx417.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx417.43" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx417.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx417.45" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx417.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Circle</span><span id="lstnumberx417.47" class="ltx_text ltx_font_typewriter">((0,0),2),</span><span id="lstnumberx417.48" class="ltx_text ltx_lst_identifier ltx_font_typewriter">red</span><span id="lstnumberx417.49" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx417.50" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx417.51" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx417.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Circle</span><span id="lstnumberx417.53" class="ltx_text ltx_font_typewriter">((0,0),3),</span><span id="lstnumberx417.54" class="ltx_text ltx_lst_identifier ltx_font_typewriter">green</span><span id="lstnumberx417.55" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx417.56" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx417.57" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx417.58" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Circle</span><span id="lstnumberx417.59" class="ltx_text ltx_font_typewriter">((0,0),4),</span><span id="lstnumberx417.60" class="ltx_text ltx_lst_identifier ltx_font_typewriter">blue</span><span id="lstnumberx417.61" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx417.62" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx417.63" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx417.64" class="ltx_text ltx_lst_identifier ltx_font_typewriter">zone</span><span id="lstnumberx417.65" class="ltx_text ltx_font_typewriter">--</span><span id="lstnumberx417.66" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ztwo</span><span id="lstnumberx417.67" class="ltx_text ltx_font_typewriter">--</span><span id="lstnumberx417.68" class="ltx_text ltx_lst_identifier ltx_font_typewriter">zthree</span><span id="lstnumberx417.69" class="ltx_text ltx_font_typewriter">--</span><span id="lstnumberx417.70" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cycle</span><span id="lstnumberx417.71" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx417.72" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx417.73" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx417.74" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndot</span><span id="lstnumberx417.75" class="ltx_text ltx_font_typewriter">("</span><span id="lstnumberx417.76" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_1$</span><span id="lstnumberx417.77" class="ltx_text ltx_font_typewriter">",</span><span id="lstnumberx417.78" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.79" class="ltx_text ltx_lst_identifier ltx_font_typewriter">zone</span><span id="lstnumberx417.80" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx417.81" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.82" class="ltx_text ltx_lst_identifier ltx_font_typewriter">E</span><span id="lstnumberx417.83" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx417.84" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndot</span><span id="lstnumberx417.85" class="ltx_text ltx_font_typewriter">("</span><span id="lstnumberx417.86" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_2$</span><span id="lstnumberx417.87" class="ltx_text ltx_font_typewriter">",</span><span id="lstnumberx417.88" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.89" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ztwo</span><span id="lstnumberx417.90" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx417.91" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.92" class="ltx_text ltx_lst_identifier ltx_font_typewriter">N</span><span id="lstnumberx417.93" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx417.94" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndot</span><span id="lstnumberx417.95" class="ltx_text ltx_font_typewriter">("</span><span id="lstnumberx417.96" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_3$</span><span id="lstnumberx417.97" class="ltx_text ltx_font_typewriter">",</span><span id="lstnumberx417.98" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.99" class="ltx_text ltx_lst_identifier ltx_font_typewriter">zthree</span><span id="lstnumberx417.100" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx417.101" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.102" class="ltx_text ltx_lst_identifier ltx_font_typewriter">SW</span><span id="lstnumberx417.103" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx417.104" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx417.105" class="ltx_text ltx_font_typewriter">[/</span><span id="lstnumberx417.106" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asy</span><span id="lstnumberx417.107" class="ltx_text ltx_font_typewriter">]\</span><span id="lstnumberx417.108" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx417.109" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx417.110" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nAlternative</span><span id="lstnumberx417.111" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx417.112" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.113" class="ltx_text ltx_lst_identifier ltx_font_typewriter">For</span><span id="lstnumberx417.114" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.115" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equality</span><span id="lstnumberx417.116" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.117" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx417.118" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.119" class="ltx_text ltx_lst_identifier ltx_font_typewriter">occur</span><span id="lstnumberx417.120" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx417.121" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.122" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx417.123" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.124" class="ltx_text ltx_lst_identifier ltx_font_typewriter">must</span><span id="lstnumberx417.125" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.126" class="ltx_text ltx_lst_identifier ltx_font_typewriter">have</span><span id="lstnumberx417.127" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.128" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_1</span><span id="lstnumberx417.129" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.130" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx417.131" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.132" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx417.133" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.134" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx417.135" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.136" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx417.137" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.138" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx417.139" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.140" class="ltx_text ltx_font_typewriter">0.</span><span id="lstnumberx417.141" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx417.142" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx417.143" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Without</span><span id="lstnumberx417.144" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.145" class="ltx_text ltx_lst_identifier ltx_font_typewriter">loss</span><span id="lstnumberx417.146" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.147" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx417.148" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.149" class="ltx_text ltx_lst_identifier ltx_font_typewriter">generality</span><span id="lstnumberx417.150" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx417.151" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.152" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx417.153" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.154" class="ltx_text ltx_lst_identifier ltx_font_typewriter">can</span><span id="lstnumberx417.155" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.156" class="ltx_text ltx_lst_identifier ltx_font_typewriter">assume</span><span id="lstnumberx417.157" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.158" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx417.159" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.160" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_1</span><span id="lstnumberx417.161" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.162" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx417.163" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.164" class="ltx_text ltx_font_typewriter">2.</span><span id="lstnumberx417.165" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx417.166" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx417.167" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Then</span><span id="lstnumberx417.168" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.169" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_2</span><span id="lstnumberx417.170" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.171" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx417.172" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.173" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx417.174" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.175" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx417.176" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.177" class="ltx_text ltx_font_typewriter">-2.</span><span id="lstnumberx417.178" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx417.179" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx417.180" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Let</span><span id="lstnumberx417.181" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.182" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_2</span><span id="lstnumberx417.183" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.184" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx417.185" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.186" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx417.187" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.188" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx417.189" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.190" class="ltx_text ltx_lst_identifier ltx_font_typewriter">iy$</span><span id="lstnumberx417.191" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.192" class="ltx_text ltx_lst_identifier ltx_font_typewriter">so</span><span id="lstnumberx417.193" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.194" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx417.195" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.196" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_3</span><span id="lstnumberx417.197" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.198" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx417.199" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.200" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx417.201" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx417.202" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.203" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx417.204" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.205" class="ltx_text ltx_font_typewriter">2</span><span id="lstnumberx417.206" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.207" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx417.208" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.209" class="ltx_text ltx_lst_identifier ltx_font_typewriter">iy</span><span id="lstnumberx417.210" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx417.211" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx417.212" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.213" class="ltx_text ltx_lst_identifier ltx_font_typewriter">where</span><span id="lstnumberx417.214" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.215" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x$</span><span id="lstnumberx417.216" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.217" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx417.218" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.219" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y$</span><span id="lstnumberx417.220" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.221" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx417.222" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.223" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx417.224" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.225" class="ltx_text ltx_lst_identifier ltx_font_typewriter">numbers</span><span id="lstnumberx417.226" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx417.227" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx417.228" class="ltx_text ltx_lst_identifier ltx_font_typewriter">We</span><span id="lstnumberx417.229" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.230" class="ltx_text ltx_lst_identifier ltx_font_typewriter">need</span><span id="lstnumberx417.231" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx417.232" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx417.233" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx417.234" class="ltx_text ltx_lst_identifier ltx_font_typewriter">begin</span><span id="lstnumberx417.235" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx417.236" class="ltx_text ltx_lst_identifier ltx_font_typewriter">align</span><span id="lstnumberx417.237" class="ltx_text ltx_font_typewriter">*}\</span><span id="lstnumberx417.238" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx417.239" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx417.240" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx417.241" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_2</span><span id="lstnumberx417.242" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx417.243" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.244" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx417.245" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.246" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx417.247" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx417.248" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.249" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx417.250" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.251" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx417.252" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx417.253" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.254" class="ltx_text ltx_font_typewriter">&amp;=</span><span id="lstnumberx417.255" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.256" class="ltx_text ltx_font_typewriter">9</span><span id="lstnumberx417.257" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.258" class="ltx_text ltx_font_typewriter">\\\\\</span><span id="lstnumberx417.259" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx417.260" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx417.261" class="ltx_text ltx_font_typewriter">|</span><span id="lstnumberx417.262" class="ltx_text ltx_lst_identifier ltx_font_typewriter">z_3</span><span id="lstnumberx417.263" class="ltx_text ltx_font_typewriter">|^2</span><span id="lstnumberx417.264" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.265" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx417.266" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.267" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx417.268" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx417.269" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.270" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx417.271" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.272" class="ltx_text ltx_font_typewriter">2)^2</span><span id="lstnumberx417.273" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.274" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx417.275" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.276" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx417.277" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx417.278" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.279" class="ltx_text ltx_font_typewriter">&amp;=</span><span id="lstnumberx417.280" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.281" class="ltx_text ltx_font_typewriter">16.\</span><span id="lstnumberx417.282" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx417.283" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx417.284" class="ltx_text ltx_lst_identifier ltx_font_typewriter">end</span><span id="lstnumberx417.285" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx417.286" class="ltx_text ltx_lst_identifier ltx_font_typewriter">align</span><span id="lstnumberx417.287" class="ltx_text ltx_font_typewriter">*}</span><span id="lstnumberx417.288" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Subtracting</span><span id="lstnumberx417.289" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.290" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx417.291" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.292" class="ltx_text ltx_lst_identifier ltx_font_typewriter">first</span><span id="lstnumberx417.293" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.294" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equation</span><span id="lstnumberx417.295" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.296" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx417.297" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.298" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx417.299" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.300" class="ltx_text ltx_lst_identifier ltx_font_typewriter">second</span><span id="lstnumberx417.301" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx417.302" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.303" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx417.304" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.305" class="ltx_text ltx_lst_identifier ltx_font_typewriter">get</span><span id="lstnumberx417.306" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.307" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$4x</span><span id="lstnumberx417.308" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.309" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx417.310" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.311" class="ltx_text ltx_font_typewriter">4</span><span id="lstnumberx417.312" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.313" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx417.314" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.315" class="ltx_text ltx_font_typewriter">7,</span><span id="lstnumberx417.316" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx417.317" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.318" class="ltx_text ltx_lst_identifier ltx_font_typewriter">or</span><span id="lstnumberx417.319" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.320" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x</span><span id="lstnumberx417.321" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.322" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx417.323" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.324" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx417.325" class="ltx_text ltx_lst_identifier ltx_font_typewriter">dfrac34</span><span id="lstnumberx417.326" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx417.327" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx417.328" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx417.329" class="ltx_text ltx_lst_identifier ltx_font_typewriter">One</span><span id="lstnumberx417.330" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.331" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solution</span><span id="lstnumberx417.332" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.333" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx417.334" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.335" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_2</span><span id="lstnumberx417.336" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.337" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx417.338" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.339" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx417.340" class="ltx_text ltx_lst_identifier ltx_font_typewriter">dfrac34</span><span id="lstnumberx417.341" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.342" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx417.343" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.344" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx417.345" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx417.346" class="ltx_text ltx_lst_identifier ltx_font_typewriter">dfrac</span><span id="lstnumberx417.347" class="ltx_text ltx_font_typewriter">{3\\</span><span id="lstnumberx417.348" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx417.349" class="ltx_text ltx_font_typewriter">{15}}{4}</span><span id="lstnumberx417.350" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx417.351" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.352" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx417.353" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.354" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$z_3</span><span id="lstnumberx417.355" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.356" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx417.357" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.358" class="ltx_text ltx_font_typewriter">-\\</span><span id="lstnumberx417.359" class="ltx_text ltx_lst_identifier ltx_font_typewriter">dfrac</span><span id="lstnumberx417.360" class="ltx_text ltx_font_typewriter">{11}4</span><span id="lstnumberx417.361" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.362" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx417.363" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.364" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx417.365" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx417.366" class="ltx_text ltx_lst_identifier ltx_font_typewriter">dfrac</span><span id="lstnumberx417.367" class="ltx_text ltx_font_typewriter">{3\\</span><span id="lstnumberx417.368" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx417.369" class="ltx_text ltx_font_typewriter">{15}}{4}.</span><span id="lstnumberx417.370" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx417.371" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx417.372" class="ltx_text ltx_lst_identifier ltx_font_typewriter">This</span><span id="lstnumberx417.373" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.374" class="ltx_text ltx_lst_identifier ltx_font_typewriter">example</span><span id="lstnumberx417.375" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.376" class="ltx_text ltx_lst_identifier ltx_font_typewriter">shows</span><span id="lstnumberx417.377" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.378" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx417.379" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.380" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equality</span><span id="lstnumberx417.381" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.382" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx417.383" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.384" class="ltx_text ltx_lst_identifier ltx_font_typewriter">possible</span><span id="lstnumberx417.385" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx417.386" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.387" class="ltx_text ltx_lst_identifier ltx_font_typewriter">so</span><span id="lstnumberx417.388" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.389" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx417.390" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.391" class="ltx_text ltx_lst_identifier ltx_font_typewriter">maximum</span><span id="lstnumberx417.392" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.393" class="ltx_text ltx_lst_identifier ltx_font_typewriter">value</span><span id="lstnumberx417.394" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.395" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx417.396" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx417.397" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx417.398" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx417.399" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx417.400" class="ltx_text ltx_font_typewriter">{87}.</span><span id="lstnumberx417.401" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx417.402" class="ltx_text ltx_font_typewriter">’</span>
</div>
</div>
</figure>
<figure id="LST9" class="ltx_float ltx_lstlisting">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float">Listing 9: </span>Failure case for <span id="LST9.3.1" class="ltx_text ltx_font_smallcaps">ToRA</span>: <span id="LST9.4.2" class="ltx_text ltx_font_bold">Syntax Error</span>.</figcaption>
<div id="LST9.5" class="ltx_listing ltx_lstlisting ltx_framed ltx_framed_topbottom ltx_listing">
<div class="ltx_listing_data"><a href="data:text/plain;base64,Idx: 106
Type: Algebra
Level: Level 5
Question: 'Below is a portion of the graph of a function, $y=h(x)$:\n\n[asy]\nimport graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.75,xmax=8.25,ymin=-1.25,ymax=10.25;\n\npen cqcqcq=rgb(0.75,0.75,0.75);\n\n/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1;\nfor(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);\n\nLabel laxis; laxis.p=fontsize(10);\n\nxaxis("",xmin,xmax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis("",ymin,ymax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);\nreal f1(real x){return (x-0.5)*(x-2.5)*(x-6.5)*(x-7.5)/16+x;}\ndraw(graph(f1,-0.25,8.25),linewidth(1));\nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);\nlabel("$y=h(x)$",(8.5,8),E);\n[/asy]\n\nWhat is the sum of all integers $x$ in the interval shown ($0\\le x\\le 8$) such that $h(x)>x$ ?'
Solution:
 ```python
from sympy import symbols, solve

def sum_of_x():
    """Below is a portion of the graph of a function, $y=h(x)$:

[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.75,xmax=8.25,ymin=-1.25,ymax=10.25;

pen cqcqcq=rgb(0.75,0.75,0.75);

/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1;
for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);

Label laxis; laxis.p=fontsize(10);

xaxis("",xmin,xmax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis("",ymin,ymax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);
real f1(real x){return (x-0.5)*(x-2.5)*(x-6.5)*(x-7.5)/16+x;}
draw(graph(f1,-0.25,8.25),linewidth(1));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
label("$y=h(x)$",(8.5,8),E);
[/asy]

What is the sum of all integers $x$ in the interval shown ($0\le x\le 8$) such that $h(x)>x$ ?"""
    x = symbols('x')
    h_x = (x - 0.5) * (x - 2.5) * (x - 6.5) * (x - 7.5) / 16 + x

    # Find the x-values where h(x) > x
    x_values = [x for x in range(9) if h_x.subs(x, x) > x]

    # Calculate the sum of these x-values
    sum_x = sum(x_values)

    return sum_x

result = sum_of_x()
print(result)
```
```output
@\hlerror{TypeError: cannot determine truth value of Relational}@
```
The code above has an error. The `if` statement can only be used with boolean values, but the `h_x.subs(x, x) > x` comparison is not a boolean value. We need to find a different way to solve this problem.

Let's analyze the problem. We are asked to find the sum of all integers $x$ in the interval shown ($0\le x\le 8$) such that $h(x)>x$.

We can see that $h(x)$ is a parabola that opens upwards. For a parabola that opens upwards, the maximum value of the parabola will occur at the midpoint of the interval, which is $x = \frac{0 + 8}{2} = 4$.

For $x < 4$, $h(x)$ is below the line $y = x$, so $h(x) \le x$. For $x > 4$, $h(x)$ is above the line $y = x$, so $h(x) > x$.

Therefore, we only need to find the sum of the integers $x$ in the interval $0 \le x \le 4$ such that $h(x) > x$.

Here's the corrected code:

```python
from sympy import symbols, solve

def sum_of_x():
    """
    Find the sum of all integers x in the interval shown ($0\le x\le 8$) such that h(x)>x.

    Returns:
        int: The sum of all integers x in the interval shown ($0\le x\le 8$) such that h(x)>x.
    """
    x = symbols('x')
    h_x = (x - 0.5) * (x - 2.5) * (x - 6.5) * (x - 7.5) / 16 + x

    # Find the x-values where h(x) > x and 0 <= x <= 4
    x_values = [x for x in range(5) if h_x.subs(x, x) > x]

    # Calculate the sum of these x-values
    sum_x = sum(x_values)

    return sum_x

result = sum_of_x()
print(result)
```
```output
@\hlerror{SyntaxError: invalid decimal literal}@
```
The code above still has an error. The `h_x.subs(x, x) > x` comparison is not a boolean value. We need to find a different way to solve this problem.

Let's analyze the problem. We are asked to find the sum of all integers $x$ in the interval shown ($0\le x\le 8$) such that $h(x)>x$.

We can see that $h(x)$ is a parabola that opens upwards. For a parabola that opens upwards, the maximum value of the parabola will occur at the midpoint of the interval, which is $x = \frac{0 + 8}{2} = 4$.

For $x < 4$, $h(x)$ is below the line $y = x$, so $h(x) \le x$. For $x > 4$, $h(x)$ is above the line $y = x$, so $h(x) > x$.

Therefore, we only need to find the sum of the integers $x$ in the interval $0 \le x \le 4$ such that $h(x) > x$.

...

Here's the corrected code:

```python
def sum_of_x():
    """
    Find the sum of all integers x in the interval shown ($0\le x\le 8$) such that h(x)>x.

    Returns:
        int: The sum of all integers x in the interval shown ($0\le x\le 8$) such that h(x)>x.
    """
    # Define the interval
    interval = range(5)

    # Find the x-values where h(x) > x
    x_values = [x for x in interval if x * (x - 4) * (x - 6) * (x - 7) / 16 > x]

    # Calculate the sum of these x-values
    sum_x = sum(x_values)

    return sum_x

result = sum_of_x()
print(result)
```
```output
@\hlerror{SyntaxError: invalid decimal literal}@
```
Reach max function call limit.

Execution: None
Prediction: 'SyntaxError:invaliddecimalliteral'
Ground Truth: '26'
Score: [False]
Reference CoT: 'We could check $h(x)$ separately for each integer $x$ from $0$ to $8$: for example, $h(0)\\approx 3.8$, so $h(0)>0$, but $h(1)\\approx -0.7$, so $h(1)\\not>1$, and so on.\n\nHowever, it is easier to see at a glance which $x$ satisfy $h(x)>x$ by superimposing a graph of $y=x$ on the graph of $y=h(x)$:\n\n[asy]\ndraw((-0.75,-0.75)--(8.25,8.25),red+1);\nimport graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.75,xmax=8.25,ymin=-1.25,ymax=10.25;\n\npen cqcqcq=rgb(0.75,0.75,0.75);\n\n/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1;\nfor(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);\n\nLabel laxis; laxis.p=fontsize(10);\n\nxaxis("",xmin,xmax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis("",ymin,ymax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);\nreal f1(real x){return (x-0.5)*(x-2.5)*(x-6.5)*(x-7.5)/16+x;}\ndraw(graph(f1,-0.25,8.25),linewidth(1));\nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);\nlabel("$y=h(x)$",(8.5,8),E);\ndot((0,0),blue); dot((3,3),blue); dot((4,4),blue); dot((5,5),blue); dot((6,6),blue); dot((8,8),blue);\n[/asy]\n\nThe six blue dots above mark the integer points $(x,x)$ which lie below the graph of $y=h(x)$, indicating that $h(x)>x$. Their $x$-coordinates are $0,3,4,5,6,8,$ which add up to $\\boxed{26}$.'" download="section/cases/failure/5_syntax_error.md">⬇</a></div>
<div id="lstnumberx418" class="ltx_listingline">
<span id="lstnumberx418.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Idx</span><span id="lstnumberx418.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx418.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx418.4" class="ltx_text ltx_font_typewriter">106</span>
</div>
<div id="lstnumberx419" class="ltx_listingline">
<span id="lstnumberx419.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Type</span><span id="lstnumberx419.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx419.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx419.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Algebra</span>
</div>
<div id="lstnumberx420" class="ltx_listingline">
<span id="lstnumberx420.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Level</span><span id="lstnumberx420.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx420.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx420.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Level</span><span id="lstnumberx420.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx420.6" class="ltx_text ltx_font_typewriter">5</span>
</div>
<div id="lstnumberx421" class="ltx_listingline">
<span id="lstnumberx421.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Question</span><span id="lstnumberx421.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx421.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.4" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx421.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Below</span><span id="lstnumberx421.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx421.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx421.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">portion</span><span id="lstnumberx421.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx421.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx421.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx421.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx421.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx421.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">function</span><span id="lstnumberx421.24" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx421.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx421.27" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx421.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h</span><span id="lstnumberx421.29" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx421.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx421.31" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx421.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx421.33" class="ltx_text ltx_font_typewriter">:\</span><span id="lstnumberx421.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx421.35" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx421.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx421.37" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx421.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asy</span><span id="lstnumberx421.39" class="ltx_text ltx_font_typewriter">]\</span><span id="lstnumberx421.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nimport</span><span id="lstnumberx421.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx421.43" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx421.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">size</span><span id="lstnumberx421.46" class="ltx_text ltx_font_typewriter">(8</span><span id="lstnumberx421.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cm</span><span id="lstnumberx421.48" class="ltx_text ltx_font_typewriter">);</span><span id="lstnumberx421.49" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.50" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx421.51" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">lsf</span><span id="lstnumberx421.53" class="ltx_text ltx_font_typewriter">=0.5;</span><span id="lstnumberx421.54" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pen</span><span id="lstnumberx421.56" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">dps</span><span id="lstnumberx421.58" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx421.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linewidth</span><span id="lstnumberx421.60" class="ltx_text ltx_font_typewriter">(0.7)+</span><span id="lstnumberx421.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">fontsize</span><span id="lstnumberx421.62" class="ltx_text ltx_font_typewriter">(10);</span><span id="lstnumberx421.63" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.64" class="ltx_text ltx_lst_identifier ltx_font_typewriter">defaultpen</span><span id="lstnumberx421.65" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx421.66" class="ltx_text ltx_lst_identifier ltx_font_typewriter">dps</span><span id="lstnumberx421.67" class="ltx_text ltx_font_typewriter">);</span><span id="lstnumberx421.68" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.69" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pen</span><span id="lstnumberx421.70" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.71" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ds</span><span id="lstnumberx421.72" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx421.73" class="ltx_text ltx_lst_identifier ltx_font_typewriter">black</span><span id="lstnumberx421.74" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx421.75" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.76" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx421.77" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.78" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx421.79" class="ltx_text ltx_font_typewriter">=-0.75,</span><span id="lstnumberx421.80" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx421.81" class="ltx_text ltx_font_typewriter">=8.25,</span><span id="lstnumberx421.82" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx421.83" class="ltx_text ltx_font_typewriter">=-1.25,</span><span id="lstnumberx421.84" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx421.85" class="ltx_text ltx_font_typewriter">=10.25;\</span><span id="lstnumberx421.86" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx421.87" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx421.88" class="ltx_text ltx_lst_identifier ltx_font_typewriter">npen</span><span id="lstnumberx421.89" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.90" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cqcqcq</span><span id="lstnumberx421.91" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx421.92" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rgb</span><span id="lstnumberx421.93" class="ltx_text ltx_font_typewriter">(0.75,0.75,0.75);\</span><span id="lstnumberx421.94" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx421.95" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx421.96" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx421.97" class="ltx_text ltx_font_typewriter">/*</span><span id="lstnumberx421.98" class="ltx_text ltx_lst_identifier ltx_font_typewriter">grid</span><span id="lstnumberx421.99" class="ltx_text ltx_font_typewriter">*/</span><span id="lstnumberx421.100" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.101" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pen</span><span id="lstnumberx421.102" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.103" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gs</span><span id="lstnumberx421.104" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx421.105" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linewidth</span><span id="lstnumberx421.106" class="ltx_text ltx_font_typewriter">(0.7)+</span><span id="lstnumberx421.107" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cqcqcq</span><span id="lstnumberx421.108" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx421.109" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linetype</span><span id="lstnumberx421.110" class="ltx_text ltx_font_typewriter">("2</span><span id="lstnumberx421.111" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.112" class="ltx_text ltx_font_typewriter">2");</span><span id="lstnumberx421.113" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.114" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx421.115" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.116" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx421.117" class="ltx_text ltx_font_typewriter">=1,</span><span id="lstnumberx421.118" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx421.119" class="ltx_text ltx_font_typewriter">=1;\</span><span id="lstnumberx421.120" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nfor</span><span id="lstnumberx421.121" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx421.122" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx421.123" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.124" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx421.125" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx421.126" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ceil</span><span id="lstnumberx421.127" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx421.128" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx421.129" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx421.130" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx421.131" class="ltx_text ltx_font_typewriter">)*</span><span id="lstnumberx421.132" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx421.133" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx421.134" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx421.135" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx421.136" class="ltx_text ltx_lst_identifier ltx_font_typewriter">floor</span><span id="lstnumberx421.137" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx421.138" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx421.139" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx421.140" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx421.141" class="ltx_text ltx_font_typewriter">)*</span><span id="lstnumberx421.142" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx421.143" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx421.144" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx421.145" class="ltx_text ltx_font_typewriter">+=</span><span id="lstnumberx421.146" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx421.147" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx421.148" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.149" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx421.150" class="ltx_text ltx_font_typewriter">((</span><span id="lstnumberx421.151" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx421.152" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx421.153" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx421.154" class="ltx_text ltx_font_typewriter">)--(</span><span id="lstnumberx421.155" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx421.156" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx421.157" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx421.158" class="ltx_text ltx_font_typewriter">),</span><span id="lstnumberx421.159" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gs</span><span id="lstnumberx421.160" class="ltx_text ltx_font_typewriter">);</span><span id="lstnumberx421.161" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.162" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx421.163" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx421.164" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx421.165" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.166" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx421.167" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx421.168" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ceil</span><span id="lstnumberx421.169" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx421.170" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx421.171" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx421.172" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx421.173" class="ltx_text ltx_font_typewriter">)*</span><span id="lstnumberx421.174" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx421.175" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx421.176" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx421.177" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx421.178" class="ltx_text ltx_lst_identifier ltx_font_typewriter">floor</span><span id="lstnumberx421.179" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx421.180" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx421.181" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx421.182" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx421.183" class="ltx_text ltx_font_typewriter">)*</span><span id="lstnumberx421.184" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx421.185" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx421.186" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx421.187" class="ltx_text ltx_font_typewriter">+=</span><span id="lstnumberx421.188" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx421.189" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx421.190" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.191" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx421.192" class="ltx_text ltx_font_typewriter">((</span><span id="lstnumberx421.193" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx421.194" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx421.195" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx421.196" class="ltx_text ltx_font_typewriter">)--(</span><span id="lstnumberx421.197" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx421.198" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx421.199" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx421.200" class="ltx_text ltx_font_typewriter">),</span><span id="lstnumberx421.201" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gs</span><span id="lstnumberx421.202" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx421.203" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx421.204" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx421.205" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nLabel</span><span id="lstnumberx421.206" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.207" class="ltx_text ltx_lst_identifier ltx_font_typewriter">laxis</span><span id="lstnumberx421.208" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx421.209" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.210" class="ltx_text ltx_lst_identifier ltx_font_typewriter">laxis</span><span id="lstnumberx421.211" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx421.212" class="ltx_text ltx_lst_identifier ltx_font_typewriter">p</span><span id="lstnumberx421.213" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx421.214" class="ltx_text ltx_lst_identifier ltx_font_typewriter">fontsize</span><span id="lstnumberx421.215" class="ltx_text ltx_font_typewriter">(10);\</span><span id="lstnumberx421.216" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx421.217" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx421.218" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nxaxis</span><span id="lstnumberx421.219" class="ltx_text ltx_font_typewriter">("",</span><span id="lstnumberx421.220" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx421.221" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx421.222" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx421.223" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx421.224" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Ticks</span><span id="lstnumberx421.225" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx421.226" class="ltx_text ltx_lst_identifier ltx_font_typewriter">laxis</span><span id="lstnumberx421.227" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx421.228" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Step</span><span id="lstnumberx421.229" class="ltx_text ltx_font_typewriter">=1.0,</span><span id="lstnumberx421.230" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Size</span><span id="lstnumberx421.231" class="ltx_text ltx_font_typewriter">=2,</span><span id="lstnumberx421.232" class="ltx_text ltx_lst_identifier ltx_font_typewriter">NoZero</span><span id="lstnumberx421.233" class="ltx_text ltx_font_typewriter">),</span><span id="lstnumberx421.234" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Arrows</span><span id="lstnumberx421.235" class="ltx_text ltx_font_typewriter">(6),</span><span id="lstnumberx421.236" class="ltx_text ltx_lst_identifier ltx_font_typewriter">above</span><span id="lstnumberx421.237" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx421.238" class="ltx_text ltx_lst_identifier ltx_font_typewriter">true</span><span id="lstnumberx421.239" class="ltx_text ltx_font_typewriter">);</span><span id="lstnumberx421.240" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.241" class="ltx_text ltx_lst_identifier ltx_font_typewriter">yaxis</span><span id="lstnumberx421.242" class="ltx_text ltx_font_typewriter">("",</span><span id="lstnumberx421.243" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx421.244" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx421.245" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx421.246" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx421.247" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Ticks</span><span id="lstnumberx421.248" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx421.249" class="ltx_text ltx_lst_identifier ltx_font_typewriter">laxis</span><span id="lstnumberx421.250" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx421.251" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Step</span><span id="lstnumberx421.252" class="ltx_text ltx_font_typewriter">=1.0,</span><span id="lstnumberx421.253" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Size</span><span id="lstnumberx421.254" class="ltx_text ltx_font_typewriter">=2,</span><span id="lstnumberx421.255" class="ltx_text ltx_lst_identifier ltx_font_typewriter">NoZero</span><span id="lstnumberx421.256" class="ltx_text ltx_font_typewriter">),</span><span id="lstnumberx421.257" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Arrows</span><span id="lstnumberx421.258" class="ltx_text ltx_font_typewriter">(6),</span><span id="lstnumberx421.259" class="ltx_text ltx_lst_identifier ltx_font_typewriter">above</span><span id="lstnumberx421.260" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx421.261" class="ltx_text ltx_lst_identifier ltx_font_typewriter">true</span><span id="lstnumberx421.262" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx421.263" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nreal</span><span id="lstnumberx421.264" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.265" class="ltx_text ltx_lst_identifier ltx_font_typewriter">f1</span><span id="lstnumberx421.266" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx421.267" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx421.268" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.269" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx421.270" class="ltx_text ltx_font_typewriter">){</span><span id="lstnumberx421.271" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx421.272" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.273" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx421.274" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx421.275" class="ltx_text ltx_font_typewriter">-0.5)*(</span><span id="lstnumberx421.276" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx421.277" class="ltx_text ltx_font_typewriter">-2.5)*(</span><span id="lstnumberx421.278" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx421.279" class="ltx_text ltx_font_typewriter">-6.5)*(</span><span id="lstnumberx421.280" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx421.281" class="ltx_text ltx_font_typewriter">-7.5)/16+</span><span id="lstnumberx421.282" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx421.283" class="ltx_text ltx_font_typewriter">;}\</span><span id="lstnumberx421.284" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx421.285" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx421.286" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx421.287" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx421.288" class="ltx_text ltx_lst_identifier ltx_font_typewriter">f1</span><span id="lstnumberx421.289" class="ltx_text ltx_font_typewriter">,-0.25,8.25),</span><span id="lstnumberx421.290" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linewidth</span><span id="lstnumberx421.291" class="ltx_text ltx_font_typewriter">(1));\</span><span id="lstnumberx421.292" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nclip</span><span id="lstnumberx421.293" class="ltx_text ltx_font_typewriter">((</span><span id="lstnumberx421.294" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx421.295" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx421.296" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx421.297" class="ltx_text ltx_font_typewriter">)--(</span><span id="lstnumberx421.298" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx421.299" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx421.300" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx421.301" class="ltx_text ltx_font_typewriter">)--(</span><span id="lstnumberx421.302" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx421.303" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx421.304" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx421.305" class="ltx_text ltx_font_typewriter">)--(</span><span id="lstnumberx421.306" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx421.307" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx421.308" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx421.309" class="ltx_text ltx_font_typewriter">)--</span><span id="lstnumberx421.310" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cycle</span><span id="lstnumberx421.311" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx421.312" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nlabel</span><span id="lstnumberx421.313" class="ltx_text ltx_font_typewriter">("</span><span id="lstnumberx421.314" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx421.315" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx421.316" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h</span><span id="lstnumberx421.317" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx421.318" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx421.319" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx421.320" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx421.321" class="ltx_text ltx_font_typewriter">",(8.5,8),</span><span id="lstnumberx421.322" class="ltx_text ltx_lst_identifier ltx_font_typewriter">E</span><span id="lstnumberx421.323" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx421.324" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx421.325" class="ltx_text ltx_font_typewriter">[/</span><span id="lstnumberx421.326" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asy</span><span id="lstnumberx421.327" class="ltx_text ltx_font_typewriter">]\</span><span id="lstnumberx421.328" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx421.329" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx421.330" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nWhat</span><span id="lstnumberx421.331" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.332" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx421.333" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.334" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx421.335" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.336" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx421.337" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.338" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx421.339" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.340" class="ltx_text ltx_lst_identifier ltx_font_typewriter">all</span><span id="lstnumberx421.341" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.342" class="ltx_text ltx_lst_identifier ltx_font_typewriter">integers</span><span id="lstnumberx421.343" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.344" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x$</span><span id="lstnumberx421.345" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.346" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx421.347" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.348" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx421.349" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.350" class="ltx_text ltx_lst_identifier ltx_font_typewriter">interval</span><span id="lstnumberx421.351" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.352" class="ltx_text ltx_lst_identifier ltx_font_typewriter">shown</span><span id="lstnumberx421.353" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.354" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx421.355" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$0</span><span id="lstnumberx421.356" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx421.357" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx421.358" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.359" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx421.360" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx421.361" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx421.362" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.363" class="ltx_text ltx_font_typewriter">8</span><span id="lstnumberx421.364" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx421.365" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx421.366" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.367" class="ltx_text ltx_lst_identifier ltx_font_typewriter">such</span><span id="lstnumberx421.368" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.369" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx421.370" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.371" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx421.372" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx421.373" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx421.374" class="ltx_text ltx_font_typewriter">)&gt;</span><span id="lstnumberx421.375" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x$</span><span id="lstnumberx421.376" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx421.377" class="ltx_text ltx_font_typewriter">?’</span>
</div>
<div id="lstnumberx422" class="ltx_listingline">
<span id="lstnumberx422.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solution</span><span id="lstnumberx422.2" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx423" class="ltx_listingline">
<span id="lstnumberx423.1" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx423.2" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx423.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span>
</div>
<div id="lstnumberx424" class="ltx_listingline">
<span id="lstnumberx424.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx424.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx424.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx424.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx424.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx424.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx424.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span><span id="lstnumberx424.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx424.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx424.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve</span>
</div>
<div id="lstnumberx425" class="ltx_listingline">
</div>
<div id="lstnumberx426" class="ltx_listingline">
<span id="lstnumberx426.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">def</span><span id="lstnumberx426.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx426.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_of_x</span><span id="lstnumberx426.4" class="ltx_text ltx_font_typewriter">():</span>
</div>
<div id="lstnumberx427" class="ltx_listingline">
<span id="lstnumberx427.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx427.2" class="ltx_text ltx_font_typewriter">"""</span><span id="lstnumberx427.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Below</span><span id="lstnumberx427.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx427.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx427.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx427.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx427.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx427.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">portion</span><span id="lstnumberx427.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx427.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx427.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx427.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx427.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx427.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx427.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx427.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx427.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx427.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx427.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx427.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">function</span><span id="lstnumberx427.22" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx427.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx427.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx427.25" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx427.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h</span><span id="lstnumberx427.27" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx427.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx427.29" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx427.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx427.31" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx428" class="ltx_listingline">
</div>
<div id="lstnumberx429" class="ltx_listingline">
<span id="lstnumberx429.1" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx429.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asy</span><span id="lstnumberx429.3" class="ltx_text ltx_font_typewriter">]</span>
</div>
<div id="lstnumberx430" class="ltx_listingline">
<span id="lstnumberx430.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx430.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx430.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx430.4" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx430.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx430.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">size</span><span id="lstnumberx430.7" class="ltx_text ltx_font_typewriter">(8</span><span id="lstnumberx430.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cm</span><span id="lstnumberx430.9" class="ltx_text ltx_font_typewriter">);</span><span id="lstnumberx430.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx430.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx430.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx430.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">lsf</span><span id="lstnumberx430.14" class="ltx_text ltx_font_typewriter">=0.5;</span><span id="lstnumberx430.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx430.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pen</span><span id="lstnumberx430.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx430.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">dps</span><span id="lstnumberx430.19" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx430.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linewidth</span><span id="lstnumberx430.21" class="ltx_text ltx_font_typewriter">(0.7)+</span><span id="lstnumberx430.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">fontsize</span><span id="lstnumberx430.23" class="ltx_text ltx_font_typewriter">(10);</span><span id="lstnumberx430.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx430.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">defaultpen</span><span id="lstnumberx430.26" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx430.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">dps</span><span id="lstnumberx430.28" class="ltx_text ltx_font_typewriter">);</span><span id="lstnumberx430.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx430.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pen</span><span id="lstnumberx430.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx430.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ds</span><span id="lstnumberx430.33" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx430.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">black</span><span id="lstnumberx430.35" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx430.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx430.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx430.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx430.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx430.40" class="ltx_text ltx_font_typewriter">=-0.75,</span><span id="lstnumberx430.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx430.42" class="ltx_text ltx_font_typewriter">=8.25,</span><span id="lstnumberx430.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx430.44" class="ltx_text ltx_font_typewriter">=-1.25,</span><span id="lstnumberx430.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx430.46" class="ltx_text ltx_font_typewriter">=10.25;</span>
</div>
<div id="lstnumberx431" class="ltx_listingline">
</div>
<div id="lstnumberx432" class="ltx_listingline">
<span id="lstnumberx432.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pen</span><span id="lstnumberx432.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx432.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cqcqcq</span><span id="lstnumberx432.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx432.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rgb</span><span id="lstnumberx432.6" class="ltx_text ltx_font_typewriter">(0.75,0.75,0.75);</span>
</div>
<div id="lstnumberx433" class="ltx_listingline">
</div>
<div id="lstnumberx434" class="ltx_listingline">
<span id="lstnumberx434.1" class="ltx_text ltx_font_typewriter">/*</span><span id="lstnumberx434.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">grid</span><span id="lstnumberx434.3" class="ltx_text ltx_font_typewriter">*/</span><span id="lstnumberx434.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx434.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pen</span><span id="lstnumberx434.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx434.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gs</span><span id="lstnumberx434.8" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx434.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linewidth</span><span id="lstnumberx434.10" class="ltx_text ltx_font_typewriter">(0.7)+</span><span id="lstnumberx434.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cqcqcq</span><span id="lstnumberx434.12" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx434.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linetype</span><span id="lstnumberx434.14" class="ltx_text ltx_font_typewriter">("2</span><span id="lstnumberx434.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx434.16" class="ltx_text ltx_font_typewriter">2");</span><span id="lstnumberx434.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx434.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx434.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx434.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx434.21" class="ltx_text ltx_font_typewriter">=1,</span><span id="lstnumberx434.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx434.23" class="ltx_text ltx_font_typewriter">=1;</span>
</div>
<div id="lstnumberx435" class="ltx_listingline">
<span id="lstnumberx435.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx435.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx435.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx435.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx435.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx435.6" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx435.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ceil</span><span id="lstnumberx435.8" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx435.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx435.10" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx435.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx435.12" class="ltx_text ltx_font_typewriter">)*</span><span id="lstnumberx435.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx435.14" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx435.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx435.16" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx435.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">floor</span><span id="lstnumberx435.18" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx435.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx435.20" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx435.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx435.22" class="ltx_text ltx_font_typewriter">)*</span><span id="lstnumberx435.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx435.24" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx435.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx435.26" class="ltx_text ltx_font_typewriter">+=</span><span id="lstnumberx435.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx435.28" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx435.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx435.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx435.31" class="ltx_text ltx_font_typewriter">((</span><span id="lstnumberx435.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx435.33" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx435.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx435.35" class="ltx_text ltx_font_typewriter">)--(</span><span id="lstnumberx435.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx435.37" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx435.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx435.39" class="ltx_text ltx_font_typewriter">),</span><span id="lstnumberx435.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gs</span><span id="lstnumberx435.41" class="ltx_text ltx_font_typewriter">);</span><span id="lstnumberx435.42" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx435.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx435.44" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx435.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx435.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx435.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx435.48" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx435.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ceil</span><span id="lstnumberx435.50" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx435.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx435.52" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx435.53" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx435.54" class="ltx_text ltx_font_typewriter">)*</span><span id="lstnumberx435.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx435.56" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx435.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx435.58" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx435.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">floor</span><span id="lstnumberx435.60" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx435.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx435.62" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx435.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx435.64" class="ltx_text ltx_font_typewriter">)*</span><span id="lstnumberx435.65" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx435.66" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx435.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx435.68" class="ltx_text ltx_font_typewriter">+=</span><span id="lstnumberx435.69" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx435.70" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx435.71" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx435.72" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx435.73" class="ltx_text ltx_font_typewriter">((</span><span id="lstnumberx435.74" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx435.75" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx435.76" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx435.77" class="ltx_text ltx_font_typewriter">)--(</span><span id="lstnumberx435.78" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx435.79" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx435.80" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx435.81" class="ltx_text ltx_font_typewriter">),</span><span id="lstnumberx435.82" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gs</span><span id="lstnumberx435.83" class="ltx_text ltx_font_typewriter">);</span>
</div>
<div id="lstnumberx436" class="ltx_listingline">
</div>
<div id="lstnumberx437" class="ltx_listingline">
<span id="lstnumberx437.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Label</span><span id="lstnumberx437.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx437.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">laxis</span><span id="lstnumberx437.4" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx437.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx437.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">laxis</span><span id="lstnumberx437.7" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx437.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">p</span><span id="lstnumberx437.9" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx437.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">fontsize</span><span id="lstnumberx437.11" class="ltx_text ltx_font_typewriter">(10);</span>
</div>
<div id="lstnumberx438" class="ltx_listingline">
</div>
<div id="lstnumberx439" class="ltx_listingline">
<span id="lstnumberx439.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xaxis</span><span id="lstnumberx439.2" class="ltx_text ltx_font_typewriter">("",</span><span id="lstnumberx439.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx439.4" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx439.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx439.6" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx439.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Ticks</span><span id="lstnumberx439.8" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx439.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">laxis</span><span id="lstnumberx439.10" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx439.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Step</span><span id="lstnumberx439.12" class="ltx_text ltx_font_typewriter">=1.0,</span><span id="lstnumberx439.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Size</span><span id="lstnumberx439.14" class="ltx_text ltx_font_typewriter">=2,</span><span id="lstnumberx439.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">NoZero</span><span id="lstnumberx439.16" class="ltx_text ltx_font_typewriter">),</span><span id="lstnumberx439.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Arrows</span><span id="lstnumberx439.18" class="ltx_text ltx_font_typewriter">(6),</span><span id="lstnumberx439.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">above</span><span id="lstnumberx439.20" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx439.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">true</span><span id="lstnumberx439.22" class="ltx_text ltx_font_typewriter">);</span><span id="lstnumberx439.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx439.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">yaxis</span><span id="lstnumberx439.25" class="ltx_text ltx_font_typewriter">("",</span><span id="lstnumberx439.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx439.27" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx439.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx439.29" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx439.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Ticks</span><span id="lstnumberx439.31" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx439.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">laxis</span><span id="lstnumberx439.33" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx439.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Step</span><span id="lstnumberx439.35" class="ltx_text ltx_font_typewriter">=1.0,</span><span id="lstnumberx439.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Size</span><span id="lstnumberx439.37" class="ltx_text ltx_font_typewriter">=2,</span><span id="lstnumberx439.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">NoZero</span><span id="lstnumberx439.39" class="ltx_text ltx_font_typewriter">),</span><span id="lstnumberx439.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Arrows</span><span id="lstnumberx439.41" class="ltx_text ltx_font_typewriter">(6),</span><span id="lstnumberx439.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">above</span><span id="lstnumberx439.43" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx439.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">true</span><span id="lstnumberx439.45" class="ltx_text ltx_font_typewriter">);</span>
</div>
<div id="lstnumberx440" class="ltx_listingline">
<span id="lstnumberx440.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx440.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx440.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">f1</span><span id="lstnumberx440.4" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx440.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx440.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx440.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx440.8" class="ltx_text ltx_font_typewriter">){</span><span id="lstnumberx440.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx440.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx440.11" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx440.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx440.13" class="ltx_text ltx_font_typewriter">-0.5)*(</span><span id="lstnumberx440.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx440.15" class="ltx_text ltx_font_typewriter">-2.5)*(</span><span id="lstnumberx440.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx440.17" class="ltx_text ltx_font_typewriter">-6.5)*(</span><span id="lstnumberx440.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx440.19" class="ltx_text ltx_font_typewriter">-7.5)/16+</span><span id="lstnumberx440.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx440.21" class="ltx_text ltx_font_typewriter">;}</span>
</div>
<div id="lstnumberx441" class="ltx_listingline">
<span id="lstnumberx441.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx441.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx441.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx441.4" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx441.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">f1</span><span id="lstnumberx441.6" class="ltx_text ltx_font_typewriter">,-0.25,8.25),</span><span id="lstnumberx441.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linewidth</span><span id="lstnumberx441.8" class="ltx_text ltx_font_typewriter">(1));</span>
</div>
<div id="lstnumberx442" class="ltx_listingline">
<span id="lstnumberx442.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">clip</span><span id="lstnumberx442.2" class="ltx_text ltx_font_typewriter">((</span><span id="lstnumberx442.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx442.4" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx442.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx442.6" class="ltx_text ltx_font_typewriter">)--(</span><span id="lstnumberx442.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx442.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx442.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx442.10" class="ltx_text ltx_font_typewriter">)--(</span><span id="lstnumberx442.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx442.12" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx442.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx442.14" class="ltx_text ltx_font_typewriter">)--(</span><span id="lstnumberx442.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx442.16" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx442.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx442.18" class="ltx_text ltx_font_typewriter">)--</span><span id="lstnumberx442.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cycle</span><span id="lstnumberx442.20" class="ltx_text ltx_font_typewriter">);</span>
</div>
<div id="lstnumberx443" class="ltx_listingline">
<span id="lstnumberx443.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">label</span><span id="lstnumberx443.2" class="ltx_text ltx_font_typewriter">("</span><span id="lstnumberx443.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx443.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx443.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h</span><span id="lstnumberx443.6" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx443.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx443.8" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx443.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx443.10" class="ltx_text ltx_font_typewriter">",(8.5,8),</span><span id="lstnumberx443.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">E</span><span id="lstnumberx443.12" class="ltx_text ltx_font_typewriter">);</span>
</div>
<div id="lstnumberx444" class="ltx_listingline">
<span id="lstnumberx444.1" class="ltx_text ltx_font_typewriter">[/</span><span id="lstnumberx444.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asy</span><span id="lstnumberx444.3" class="ltx_text ltx_font_typewriter">]</span>
</div>
<div id="lstnumberx445" class="ltx_listingline">
</div>
<div id="lstnumberx446" class="ltx_listingline">
<span id="lstnumberx446.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">What</span><span id="lstnumberx446.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx446.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx446.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx446.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx446.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">all</span><span id="lstnumberx446.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">integers</span><span id="lstnumberx446.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x$</span><span id="lstnumberx446.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx446.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx446.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">interval</span><span id="lstnumberx446.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">shown</span><span id="lstnumberx446.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.25" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx446.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$0</span><span id="lstnumberx446.27" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx446.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx446.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx446.31" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx446.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx446.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.34" class="ltx_text ltx_font_typewriter">8</span><span id="lstnumberx446.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx446.36" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx446.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">such</span><span id="lstnumberx446.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx446.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx446.43" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx446.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx446.45" class="ltx_text ltx_font_typewriter">)&gt;</span><span id="lstnumberx446.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x$</span><span id="lstnumberx446.47" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx446.48" class="ltx_text ltx_font_typewriter">?"""</span>
</div>
<div id="lstnumberx447" class="ltx_listingline">
<span id="lstnumberx447.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx447.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx447.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx447.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx447.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx447.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span><span id="lstnumberx447.7" class="ltx_text ltx_font_typewriter">(’</span><span id="lstnumberx447.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx447.9" class="ltx_text ltx_font_typewriter">’)</span>
</div>
<div id="lstnumberx448" class="ltx_listingline">
<span id="lstnumberx448.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx448.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h_x</span><span id="lstnumberx448.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx448.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.6" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx448.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx448.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.9" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx448.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.11" class="ltx_text ltx_font_typewriter">0.5)</span><span id="lstnumberx448.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.13" class="ltx_text ltx_font_typewriter">*</span><span id="lstnumberx448.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.15" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx448.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx448.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.18" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx448.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.20" class="ltx_text ltx_font_typewriter">2.5)</span><span id="lstnumberx448.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.22" class="ltx_text ltx_font_typewriter">*</span><span id="lstnumberx448.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.24" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx448.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx448.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.27" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx448.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.29" class="ltx_text ltx_font_typewriter">6.5)</span><span id="lstnumberx448.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.31" class="ltx_text ltx_font_typewriter">*</span><span id="lstnumberx448.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.33" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx448.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx448.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.36" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx448.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.38" class="ltx_text ltx_font_typewriter">7.5)</span><span id="lstnumberx448.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.40" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx448.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.42" class="ltx_text ltx_font_typewriter">16</span><span id="lstnumberx448.43" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.44" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx448.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx448.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span>
</div>
<div id="lstnumberx449" class="ltx_listingline">
</div>
<div id="lstnumberx450" class="ltx_listingline">
<span id="lstnumberx450.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx450.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx450.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx450.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Find</span><span id="lstnumberx450.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx450.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx450.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx450.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx450.9" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx450.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">values</span><span id="lstnumberx450.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx450.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">where</span><span id="lstnumberx450.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx450.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h</span><span id="lstnumberx450.15" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx450.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx450.17" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx450.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx450.19" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx450.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx450.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span>
</div>
<div id="lstnumberx451" class="ltx_listingline">
<span id="lstnumberx451.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx451.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x_values</span><span id="lstnumberx451.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx451.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx451.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx451.6" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx451.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx451.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx451.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx451.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx451.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx451.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx451.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx451.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx451.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">range</span><span id="lstnumberx451.16" class="ltx_text ltx_font_typewriter">(9)</span><span id="lstnumberx451.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx451.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">if</span><span id="lstnumberx451.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx451.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h_x</span><span id="lstnumberx451.21" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx451.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">subs</span><span id="lstnumberx451.23" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx451.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx451.25" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx451.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx451.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx451.28" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx451.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx451.30" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx451.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx451.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx451.33" class="ltx_text ltx_font_typewriter">]</span>
</div>
<div id="lstnumberx452" class="ltx_listingline">
</div>
<div id="lstnumberx453" class="ltx_listingline">
<span id="lstnumberx453.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx453.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx453.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx453.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Calculate</span><span id="lstnumberx453.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx453.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx453.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx453.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx453.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx453.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx453.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx453.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">these</span><span id="lstnumberx453.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx453.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx453.15" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx453.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">values</span>
</div>
<div id="lstnumberx454" class="ltx_listingline">
<span id="lstnumberx454.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx454.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_x</span><span id="lstnumberx454.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx454.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx454.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx454.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx454.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx454.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x_values</span><span id="lstnumberx454.9" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx455" class="ltx_listingline">
</div>
<div id="lstnumberx456" class="ltx_listingline">
<span id="lstnumberx456.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx456.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx456.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx456.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_x</span>
</div>
<div id="lstnumberx457" class="ltx_listingline">
</div>
<div id="lstnumberx458" class="ltx_listingline">
<span id="lstnumberx458.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx458.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx458.3" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx458.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx458.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_of_x</span><span id="lstnumberx458.6" class="ltx_text ltx_font_typewriter">()</span>
</div>
<div id="lstnumberx459" class="ltx_listingline">
<span id="lstnumberx459.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">print</span><span id="lstnumberx459.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx459.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx459.4" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx460" class="ltx_listingline">
<span id="lstnumberx460.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx461" class="ltx_listingline">
<span id="lstnumberx461.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx461.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span>
</div>
<div id="lstnumberx462" class="ltx_listingline">
<span id="lstnumberx462.1" class="ltx_text ltx_font_typewriter" style="background-color:#F47874;">TypeError: cannot determine truth value of Relational</span>
</div>
<div id="lstnumberx463" class="ltx_listingline">
<span id="lstnumberx463.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx464" class="ltx_listingline">
<span id="lstnumberx464.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx464.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">code</span><span id="lstnumberx464.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">above</span><span id="lstnumberx464.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">has</span><span id="lstnumberx464.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">an</span><span id="lstnumberx464.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">error</span><span id="lstnumberx464.12" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx464.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx464.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.16" class="ltx_text ltx_font_typewriter">‘</span><span id="lstnumberx464.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">if</span><span id="lstnumberx464.18" class="ltx_text ltx_font_typewriter">‘</span><span id="lstnumberx464.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">statement</span><span id="lstnumberx464.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">can</span><span id="lstnumberx464.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">only</span><span id="lstnumberx464.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">be</span><span id="lstnumberx464.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">used</span><span id="lstnumberx464.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">with</span><span id="lstnumberx464.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boolean</span><span id="lstnumberx464.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">values</span><span id="lstnumberx464.35" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx464.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">but</span><span id="lstnumberx464.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx464.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.41" class="ltx_text ltx_font_typewriter">‘</span><span id="lstnumberx464.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h_x</span><span id="lstnumberx464.43" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx464.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">subs</span><span id="lstnumberx464.45" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx464.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx464.47" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx464.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx464.50" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx464.51" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.52" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx464.53" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.54" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx464.55" class="ltx_text ltx_font_typewriter">‘</span><span id="lstnumberx464.56" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">comparison</span><span id="lstnumberx464.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx464.60" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">not</span><span id="lstnumberx464.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx464.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.65" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boolean</span><span id="lstnumberx464.66" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">value</span><span id="lstnumberx464.68" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx464.69" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.70" class="ltx_text ltx_lst_identifier ltx_font_typewriter">We</span><span id="lstnumberx464.71" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.72" class="ltx_text ltx_lst_identifier ltx_font_typewriter">need</span><span id="lstnumberx464.73" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.74" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx464.75" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.76" class="ltx_text ltx_lst_identifier ltx_font_typewriter">find</span><span id="lstnumberx464.77" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.78" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx464.79" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.80" class="ltx_text ltx_lst_identifier ltx_font_typewriter">different</span><span id="lstnumberx464.81" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.82" class="ltx_text ltx_lst_identifier ltx_font_typewriter">way</span><span id="lstnumberx464.83" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.84" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx464.85" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.86" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve</span><span id="lstnumberx464.87" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.88" class="ltx_text ltx_lst_identifier ltx_font_typewriter">this</span><span id="lstnumberx464.89" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx464.90" class="ltx_text ltx_lst_identifier ltx_font_typewriter">problem</span><span id="lstnumberx464.91" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx465" class="ltx_listingline">
</div>
<div id="lstnumberx466" class="ltx_listingline">
<span id="lstnumberx466.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Let</span><span id="lstnumberx466.2" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx466.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx466.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">analyze</span><span id="lstnumberx466.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx466.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">problem</span><span id="lstnumberx466.10" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx466.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">We</span><span id="lstnumberx466.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx466.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asked</span><span id="lstnumberx466.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx466.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">find</span><span id="lstnumberx466.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx466.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx466.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx466.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">all</span><span id="lstnumberx466.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">integers</span><span id="lstnumberx466.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x$</span><span id="lstnumberx466.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx466.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx466.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">interval</span><span id="lstnumberx466.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">shown</span><span id="lstnumberx466.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.42" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx466.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$0</span><span id="lstnumberx466.44" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx466.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx466.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx466.48" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx466.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx466.50" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.51" class="ltx_text ltx_font_typewriter">8</span><span id="lstnumberx466.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx466.53" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx466.54" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">such</span><span id="lstnumberx466.56" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx466.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx466.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx466.60" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx466.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx466.62" class="ltx_text ltx_font_typewriter">)&gt;</span><span id="lstnumberx466.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x$</span><span id="lstnumberx466.64" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx467" class="ltx_listingline">
</div>
<div id="lstnumberx468" class="ltx_listingline">
<span id="lstnumberx468.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">We</span><span id="lstnumberx468.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">can</span><span id="lstnumberx468.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">see</span><span id="lstnumberx468.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx468.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx468.10" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx468.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx468.12" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx468.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx468.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx468.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx468.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">parabola</span><span id="lstnumberx468.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx468.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">opens</span><span id="lstnumberx468.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">upwards</span><span id="lstnumberx468.26" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx468.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">For</span><span id="lstnumberx468.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx468.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">parabola</span><span id="lstnumberx468.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx468.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">opens</span><span id="lstnumberx468.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">upwards</span><span id="lstnumberx468.39" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx468.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx468.42" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">maximum</span><span id="lstnumberx468.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">value</span><span id="lstnumberx468.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx468.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx468.50" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">parabola</span><span id="lstnumberx468.52" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.53" class="ltx_text ltx_lst_identifier ltx_font_typewriter">will</span><span id="lstnumberx468.54" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">occur</span><span id="lstnumberx468.56" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">at</span><span id="lstnumberx468.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx468.60" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">midpoint</span><span id="lstnumberx468.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx468.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.65" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx468.66" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">interval</span><span id="lstnumberx468.68" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx468.69" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.70" class="ltx_text ltx_lst_identifier ltx_font_typewriter">which</span><span id="lstnumberx468.71" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.72" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx468.73" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.74" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x</span><span id="lstnumberx468.75" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.76" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx468.77" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.78" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx468.79" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx468.80" class="ltx_text ltx_font_typewriter">{0</span><span id="lstnumberx468.81" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.82" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx468.83" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.84" class="ltx_text ltx_font_typewriter">8}{2}</span><span id="lstnumberx468.85" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.86" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx468.87" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx468.88" class="ltx_text ltx_font_typewriter">4</span><span id="lstnumberx468.89" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx468.90" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx469" class="ltx_listingline">
</div>
<div id="lstnumberx470" class="ltx_listingline">
<span id="lstnumberx470.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">For</span><span id="lstnumberx470.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x</span><span id="lstnumberx470.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.5" class="ltx_text ltx_font_typewriter">&lt;</span><span id="lstnumberx470.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.7" class="ltx_text ltx_font_typewriter">4</span><span id="lstnumberx470.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx470.9" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx470.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx470.12" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx470.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx470.14" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx470.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx470.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx470.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">below</span><span id="lstnumberx470.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx470.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">line</span><span id="lstnumberx470.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx470.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.27" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx470.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x$</span><span id="lstnumberx470.30" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx470.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">so</span><span id="lstnumberx470.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx470.35" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx470.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx470.37" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx470.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.39" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx470.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx470.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x$</span><span id="lstnumberx470.43" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx470.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">For</span><span id="lstnumberx470.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x</span><span id="lstnumberx470.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.49" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx470.50" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.51" class="ltx_text ltx_font_typewriter">4</span><span id="lstnumberx470.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx470.53" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx470.54" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx470.56" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx470.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx470.58" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx470.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx470.60" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx470.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">above</span><span id="lstnumberx470.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.65" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx470.66" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">line</span><span id="lstnumberx470.68" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.69" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx470.70" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.71" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx470.72" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.73" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x$</span><span id="lstnumberx470.74" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx470.75" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.76" class="ltx_text ltx_lst_identifier ltx_font_typewriter">so</span><span id="lstnumberx470.77" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.78" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx470.79" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx470.80" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx470.81" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx470.82" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.83" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx470.84" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx470.85" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x$</span><span id="lstnumberx470.86" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx471" class="ltx_listingline">
</div>
<div id="lstnumberx472" class="ltx_listingline">
<span id="lstnumberx472.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Therefore</span><span id="lstnumberx472.2" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx472.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx472.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">only</span><span id="lstnumberx472.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">need</span><span id="lstnumberx472.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx472.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">find</span><span id="lstnumberx472.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx472.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx472.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx472.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx472.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">integers</span><span id="lstnumberx472.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x$</span><span id="lstnumberx472.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx472.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx472.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">interval</span><span id="lstnumberx472.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$0</span><span id="lstnumberx472.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.34" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx472.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx472.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx472.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.39" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx472.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx472.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.42" class="ltx_text ltx_font_typewriter">4</span><span id="lstnumberx472.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx472.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">such</span><span id="lstnumberx472.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx472.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx472.50" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx472.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx472.52" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx472.53" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.54" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx472.55" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx472.56" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x$</span><span id="lstnumberx472.57" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx473" class="ltx_listingline">
</div>
<div id="lstnumberx474" class="ltx_listingline">
<span id="lstnumberx474.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Here</span><span id="lstnumberx474.2" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx474.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx474.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx474.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx474.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx474.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">corrected</span><span id="lstnumberx474.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx474.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">code</span><span id="lstnumberx474.10" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx475" class="ltx_listingline">
</div>
<div id="lstnumberx476" class="ltx_listingline">
<span id="lstnumberx476.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx476.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span>
</div>
<div id="lstnumberx477" class="ltx_listingline">
<span id="lstnumberx477.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx477.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx477.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx477.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx477.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx477.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx477.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span><span id="lstnumberx477.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx477.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx477.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve</span>
</div>
<div id="lstnumberx478" class="ltx_listingline">
</div>
<div id="lstnumberx479" class="ltx_listingline">
<span id="lstnumberx479.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">def</span><span id="lstnumberx479.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx479.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_of_x</span><span id="lstnumberx479.4" class="ltx_text ltx_font_typewriter">():</span>
</div>
<div id="lstnumberx480" class="ltx_listingline">
<span id="lstnumberx480.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx480.2" class="ltx_text ltx_font_typewriter">"""</span>
</div>
<div id="lstnumberx481" class="ltx_listingline">
<span id="lstnumberx481.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx481.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Find</span><span id="lstnumberx481.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx481.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx481.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx481.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx481.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx481.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx481.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx481.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">all</span><span id="lstnumberx481.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx481.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">integers</span><span id="lstnumberx481.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx481.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx481.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx481.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx481.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx481.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx481.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx481.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">interval</span><span id="lstnumberx481.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx481.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">shown</span><span id="lstnumberx481.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx481.24" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx481.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$0</span><span id="lstnumberx481.26" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx481.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx481.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx481.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx481.30" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx481.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx481.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx481.33" class="ltx_text ltx_font_typewriter">8</span><span id="lstnumberx481.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx481.35" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx481.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx481.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">such</span><span id="lstnumberx481.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx481.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx481.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx481.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h</span><span id="lstnumberx481.42" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx481.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx481.44" class="ltx_text ltx_font_typewriter">)&gt;</span><span id="lstnumberx481.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx481.46" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx482" class="ltx_listingline">
</div>
<div id="lstnumberx483" class="ltx_listingline">
<span id="lstnumberx483.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx483.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Returns</span><span id="lstnumberx483.3" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx484" class="ltx_listingline">
<span id="lstnumberx484.1" class="ltx_text ltx_lst_space ltx_font_typewriter">        </span><span id="lstnumberx484.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">int</span><span id="lstnumberx484.3" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx484.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx484.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx484.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx484.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx484.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx484.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx484.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx484.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">all</span><span id="lstnumberx484.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx484.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">integers</span><span id="lstnumberx484.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx484.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx484.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx484.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx484.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx484.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx484.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx484.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">interval</span><span id="lstnumberx484.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx484.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">shown</span><span id="lstnumberx484.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx484.25" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx484.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$0</span><span id="lstnumberx484.27" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx484.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx484.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx484.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx484.31" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx484.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx484.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx484.34" class="ltx_text ltx_font_typewriter">8</span><span id="lstnumberx484.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx484.36" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx484.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx484.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">such</span><span id="lstnumberx484.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx484.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx484.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx484.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h</span><span id="lstnumberx484.43" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx484.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx484.45" class="ltx_text ltx_font_typewriter">)&gt;</span><span id="lstnumberx484.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx484.47" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx485" class="ltx_listingline">
<span id="lstnumberx485.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx485.2" class="ltx_text ltx_font_typewriter">"""</span>
</div>
<div id="lstnumberx486" class="ltx_listingline">
<span id="lstnumberx486.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx486.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx486.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx486.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx486.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx486.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span><span id="lstnumberx486.7" class="ltx_text ltx_font_typewriter">(’</span><span id="lstnumberx486.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx486.9" class="ltx_text ltx_font_typewriter">’)</span>
</div>
<div id="lstnumberx487" class="ltx_listingline">
<span id="lstnumberx487.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx487.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h_x</span><span id="lstnumberx487.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx487.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.6" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx487.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx487.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.9" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx487.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.11" class="ltx_text ltx_font_typewriter">0.5)</span><span id="lstnumberx487.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.13" class="ltx_text ltx_font_typewriter">*</span><span id="lstnumberx487.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.15" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx487.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx487.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.18" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx487.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.20" class="ltx_text ltx_font_typewriter">2.5)</span><span id="lstnumberx487.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.22" class="ltx_text ltx_font_typewriter">*</span><span id="lstnumberx487.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.24" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx487.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx487.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.27" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx487.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.29" class="ltx_text ltx_font_typewriter">6.5)</span><span id="lstnumberx487.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.31" class="ltx_text ltx_font_typewriter">*</span><span id="lstnumberx487.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.33" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx487.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx487.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.36" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx487.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.38" class="ltx_text ltx_font_typewriter">7.5)</span><span id="lstnumberx487.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.40" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx487.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.42" class="ltx_text ltx_font_typewriter">16</span><span id="lstnumberx487.43" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.44" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx487.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx487.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span>
</div>
<div id="lstnumberx488" class="ltx_listingline">
</div>
<div id="lstnumberx489" class="ltx_listingline">
<span id="lstnumberx489.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx489.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx489.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx489.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Find</span><span id="lstnumberx489.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx489.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx489.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx489.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx489.9" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx489.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">values</span><span id="lstnumberx489.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx489.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">where</span><span id="lstnumberx489.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx489.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h</span><span id="lstnumberx489.15" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx489.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx489.17" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx489.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx489.19" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx489.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx489.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx489.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx489.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx489.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx489.25" class="ltx_text ltx_font_typewriter">0</span><span id="lstnumberx489.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx489.27" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx489.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx489.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx489.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx489.31" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx489.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx489.33" class="ltx_text ltx_font_typewriter">4</span>
</div>
<div id="lstnumberx490" class="ltx_listingline">
<span id="lstnumberx490.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx490.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x_values</span><span id="lstnumberx490.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx490.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx490.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx490.6" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx490.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx490.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx490.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx490.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx490.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx490.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx490.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx490.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx490.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">range</span><span id="lstnumberx490.16" class="ltx_text ltx_font_typewriter">(5)</span><span id="lstnumberx490.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx490.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">if</span><span id="lstnumberx490.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx490.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h_x</span><span id="lstnumberx490.21" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx490.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">subs</span><span id="lstnumberx490.23" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx490.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx490.25" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx490.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx490.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx490.28" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx490.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx490.30" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx490.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx490.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx490.33" class="ltx_text ltx_font_typewriter">]</span>
</div>
<div id="lstnumberx491" class="ltx_listingline">
</div>
<div id="lstnumberx492" class="ltx_listingline">
<span id="lstnumberx492.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx492.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx492.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx492.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Calculate</span><span id="lstnumberx492.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx492.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx492.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx492.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx492.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx492.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx492.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx492.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">these</span><span id="lstnumberx492.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx492.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx492.15" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx492.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">values</span>
</div>
<div id="lstnumberx493" class="ltx_listingline">
<span id="lstnumberx493.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx493.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_x</span><span id="lstnumberx493.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx493.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx493.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx493.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx493.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx493.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x_values</span><span id="lstnumberx493.9" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx494" class="ltx_listingline">
</div>
<div id="lstnumberx495" class="ltx_listingline">
<span id="lstnumberx495.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx495.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx495.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx495.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_x</span>
</div>
<div id="lstnumberx496" class="ltx_listingline">
</div>
<div id="lstnumberx497" class="ltx_listingline">
<span id="lstnumberx497.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx497.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx497.3" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx497.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx497.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_of_x</span><span id="lstnumberx497.6" class="ltx_text ltx_font_typewriter">()</span>
</div>
<div id="lstnumberx498" class="ltx_listingline">
<span id="lstnumberx498.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">print</span><span id="lstnumberx498.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx498.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx498.4" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx499" class="ltx_listingline">
<span id="lstnumberx499.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx500" class="ltx_listingline">
<span id="lstnumberx500.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx500.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span>
</div>
<div id="lstnumberx501" class="ltx_listingline">
<span id="lstnumberx501.1" class="ltx_text ltx_font_typewriter" style="background-color:#F47874;">SyntaxError: invalid decimal literal</span>
</div>
<div id="lstnumberx502" class="ltx_listingline">
<span id="lstnumberx502.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx503" class="ltx_listingline">
<span id="lstnumberx503.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx503.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">code</span><span id="lstnumberx503.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">above</span><span id="lstnumberx503.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">still</span><span id="lstnumberx503.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">has</span><span id="lstnumberx503.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">an</span><span id="lstnumberx503.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">error</span><span id="lstnumberx503.14" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx503.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx503.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.18" class="ltx_text ltx_font_typewriter">‘</span><span id="lstnumberx503.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h_x</span><span id="lstnumberx503.20" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx503.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">subs</span><span id="lstnumberx503.22" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx503.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx503.24" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx503.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx503.27" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx503.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.29" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx503.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx503.32" class="ltx_text ltx_font_typewriter">‘</span><span id="lstnumberx503.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">comparison</span><span id="lstnumberx503.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx503.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">not</span><span id="lstnumberx503.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx503.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boolean</span><span id="lstnumberx503.43" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">value</span><span id="lstnumberx503.45" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx503.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">We</span><span id="lstnumberx503.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">need</span><span id="lstnumberx503.50" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx503.52" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.53" class="ltx_text ltx_lst_identifier ltx_font_typewriter">find</span><span id="lstnumberx503.54" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx503.56" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">different</span><span id="lstnumberx503.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">way</span><span id="lstnumberx503.60" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx503.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve</span><span id="lstnumberx503.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.65" class="ltx_text ltx_lst_identifier ltx_font_typewriter">this</span><span id="lstnumberx503.66" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx503.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">problem</span><span id="lstnumberx503.68" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx504" class="ltx_listingline">
</div>
<div id="lstnumberx505" class="ltx_listingline">
<span id="lstnumberx505.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Let</span><span id="lstnumberx505.2" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx505.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx505.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">analyze</span><span id="lstnumberx505.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx505.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">problem</span><span id="lstnumberx505.10" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx505.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">We</span><span id="lstnumberx505.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx505.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asked</span><span id="lstnumberx505.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx505.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">find</span><span id="lstnumberx505.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx505.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx505.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx505.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">all</span><span id="lstnumberx505.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">integers</span><span id="lstnumberx505.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x$</span><span id="lstnumberx505.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx505.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx505.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">interval</span><span id="lstnumberx505.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">shown</span><span id="lstnumberx505.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.42" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx505.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$0</span><span id="lstnumberx505.44" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx505.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx505.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx505.48" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx505.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx505.50" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.51" class="ltx_text ltx_font_typewriter">8</span><span id="lstnumberx505.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx505.53" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx505.54" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">such</span><span id="lstnumberx505.56" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx505.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx505.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx505.60" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx505.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx505.62" class="ltx_text ltx_font_typewriter">)&gt;</span><span id="lstnumberx505.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x$</span><span id="lstnumberx505.64" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx506" class="ltx_listingline">
</div>
<div id="lstnumberx507" class="ltx_listingline">
<span id="lstnumberx507.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">We</span><span id="lstnumberx507.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">can</span><span id="lstnumberx507.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">see</span><span id="lstnumberx507.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx507.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx507.10" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx507.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx507.12" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx507.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx507.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx507.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx507.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">parabola</span><span id="lstnumberx507.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx507.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">opens</span><span id="lstnumberx507.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">upwards</span><span id="lstnumberx507.26" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx507.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">For</span><span id="lstnumberx507.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx507.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">parabola</span><span id="lstnumberx507.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx507.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">opens</span><span id="lstnumberx507.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">upwards</span><span id="lstnumberx507.39" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx507.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx507.42" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">maximum</span><span id="lstnumberx507.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">value</span><span id="lstnumberx507.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx507.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx507.50" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">parabola</span><span id="lstnumberx507.52" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.53" class="ltx_text ltx_lst_identifier ltx_font_typewriter">will</span><span id="lstnumberx507.54" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">occur</span><span id="lstnumberx507.56" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">at</span><span id="lstnumberx507.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx507.60" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">midpoint</span><span id="lstnumberx507.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx507.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.65" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx507.66" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">interval</span><span id="lstnumberx507.68" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx507.69" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.70" class="ltx_text ltx_lst_identifier ltx_font_typewriter">which</span><span id="lstnumberx507.71" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.72" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx507.73" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.74" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x</span><span id="lstnumberx507.75" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.76" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx507.77" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.78" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx507.79" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx507.80" class="ltx_text ltx_font_typewriter">{0</span><span id="lstnumberx507.81" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.82" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx507.83" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.84" class="ltx_text ltx_font_typewriter">8}{2}</span><span id="lstnumberx507.85" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.86" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx507.87" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx507.88" class="ltx_text ltx_font_typewriter">4</span><span id="lstnumberx507.89" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx507.90" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx508" class="ltx_listingline">
</div>
<div id="lstnumberx509" class="ltx_listingline">
<span id="lstnumberx509.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">For</span><span id="lstnumberx509.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x</span><span id="lstnumberx509.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.5" class="ltx_text ltx_font_typewriter">&lt;</span><span id="lstnumberx509.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.7" class="ltx_text ltx_font_typewriter">4</span><span id="lstnumberx509.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx509.9" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx509.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx509.12" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx509.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx509.14" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx509.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx509.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx509.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">below</span><span id="lstnumberx509.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx509.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">line</span><span id="lstnumberx509.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx509.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.27" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx509.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x$</span><span id="lstnumberx509.30" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx509.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">so</span><span id="lstnumberx509.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx509.35" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx509.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx509.37" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx509.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.39" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx509.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx509.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x$</span><span id="lstnumberx509.43" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx509.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">For</span><span id="lstnumberx509.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x</span><span id="lstnumberx509.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.49" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx509.50" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.51" class="ltx_text ltx_font_typewriter">4</span><span id="lstnumberx509.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx509.53" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx509.54" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx509.56" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx509.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx509.58" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx509.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx509.60" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx509.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">above</span><span id="lstnumberx509.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.65" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx509.66" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">line</span><span id="lstnumberx509.68" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.69" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx509.70" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.71" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx509.72" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.73" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x$</span><span id="lstnumberx509.74" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx509.75" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.76" class="ltx_text ltx_lst_identifier ltx_font_typewriter">so</span><span id="lstnumberx509.77" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.78" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx509.79" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx509.80" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx509.81" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx509.82" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.83" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx509.84" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx509.85" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x$</span><span id="lstnumberx509.86" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx510" class="ltx_listingline">
</div>
<div id="lstnumberx511" class="ltx_listingline">
<span id="lstnumberx511.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Therefore</span><span id="lstnumberx511.2" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx511.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx511.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">only</span><span id="lstnumberx511.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">need</span><span id="lstnumberx511.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx511.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">find</span><span id="lstnumberx511.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx511.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx511.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx511.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx511.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">integers</span><span id="lstnumberx511.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x$</span><span id="lstnumberx511.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx511.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx511.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">interval</span><span id="lstnumberx511.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$0</span><span id="lstnumberx511.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.34" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx511.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx511.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx511.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.39" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx511.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx511.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.42" class="ltx_text ltx_font_typewriter">4</span><span id="lstnumberx511.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx511.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">such</span><span id="lstnumberx511.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx511.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx511.50" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx511.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx511.52" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx511.53" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.54" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx511.55" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx511.56" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x$</span><span id="lstnumberx511.57" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx512" class="ltx_listingline">
</div>
<div id="lstnumberx513" class="ltx_listingline">
<span id="lstnumberx513.1" class="ltx_text ltx_font_typewriter">...</span>
</div>
<div id="lstnumberx514" class="ltx_listingline">
</div>
<div id="lstnumberx515" class="ltx_listingline">
<span id="lstnumberx515.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Here</span><span id="lstnumberx515.2" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx515.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx515.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx515.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx515.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx515.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">corrected</span><span id="lstnumberx515.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx515.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">code</span><span id="lstnumberx515.10" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx516" class="ltx_listingline">
</div>
<div id="lstnumberx517" class="ltx_listingline">
<span id="lstnumberx517.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx517.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span>
</div>
<div id="lstnumberx518" class="ltx_listingline">
<span id="lstnumberx518.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">def</span><span id="lstnumberx518.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx518.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_of_x</span><span id="lstnumberx518.4" class="ltx_text ltx_font_typewriter">():</span>
</div>
<div id="lstnumberx519" class="ltx_listingline">
<span id="lstnumberx519.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx519.2" class="ltx_text ltx_font_typewriter">"""</span>
</div>
<div id="lstnumberx520" class="ltx_listingline">
<span id="lstnumberx520.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx520.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Find</span><span id="lstnumberx520.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx520.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx520.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx520.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx520.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx520.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx520.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx520.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">all</span><span id="lstnumberx520.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx520.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">integers</span><span id="lstnumberx520.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx520.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx520.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx520.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx520.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx520.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx520.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx520.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">interval</span><span id="lstnumberx520.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx520.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">shown</span><span id="lstnumberx520.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx520.24" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx520.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$0</span><span id="lstnumberx520.26" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx520.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx520.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx520.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx520.30" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx520.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx520.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx520.33" class="ltx_text ltx_font_typewriter">8</span><span id="lstnumberx520.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx520.35" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx520.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx520.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">such</span><span id="lstnumberx520.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx520.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx520.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx520.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h</span><span id="lstnumberx520.42" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx520.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx520.44" class="ltx_text ltx_font_typewriter">)&gt;</span><span id="lstnumberx520.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx520.46" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx521" class="ltx_listingline">
</div>
<div id="lstnumberx522" class="ltx_listingline">
<span id="lstnumberx522.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx522.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Returns</span><span id="lstnumberx522.3" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx523" class="ltx_listingline">
<span id="lstnumberx523.1" class="ltx_text ltx_lst_space ltx_font_typewriter">        </span><span id="lstnumberx523.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">int</span><span id="lstnumberx523.3" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx523.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx523.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx523.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx523.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx523.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx523.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx523.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx523.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">all</span><span id="lstnumberx523.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx523.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">integers</span><span id="lstnumberx523.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx523.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx523.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx523.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx523.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx523.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx523.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx523.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">interval</span><span id="lstnumberx523.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx523.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">shown</span><span id="lstnumberx523.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx523.25" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx523.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$0</span><span id="lstnumberx523.27" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx523.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx523.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx523.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx523.31" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx523.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">le</span><span id="lstnumberx523.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx523.34" class="ltx_text ltx_font_typewriter">8</span><span id="lstnumberx523.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx523.36" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx523.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx523.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">such</span><span id="lstnumberx523.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx523.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx523.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx523.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h</span><span id="lstnumberx523.43" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx523.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx523.45" class="ltx_text ltx_font_typewriter">)&gt;</span><span id="lstnumberx523.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx523.47" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx524" class="ltx_listingline">
<span id="lstnumberx524.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx524.2" class="ltx_text ltx_font_typewriter">"""</span>
</div>
<div id="lstnumberx525" class="ltx_listingline">
<span id="lstnumberx525.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx525.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx525.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx525.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Define</span><span id="lstnumberx525.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx525.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx525.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx525.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">interval</span>
</div>
<div id="lstnumberx526" class="ltx_listingline">
<span id="lstnumberx526.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx526.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">interval</span><span id="lstnumberx526.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx526.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx526.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx526.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">range</span><span id="lstnumberx526.7" class="ltx_text ltx_font_typewriter">(5)</span>
</div>
<div id="lstnumberx527" class="ltx_listingline">
</div>
<div id="lstnumberx528" class="ltx_listingline">
<span id="lstnumberx528.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx528.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx528.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx528.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Find</span><span id="lstnumberx528.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx528.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx528.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx528.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx528.9" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx528.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">values</span><span id="lstnumberx528.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx528.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">where</span><span id="lstnumberx528.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx528.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h</span><span id="lstnumberx528.15" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx528.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx528.17" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx528.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx528.19" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx528.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx528.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span>
</div>
<div id="lstnumberx529" class="ltx_listingline">
<span id="lstnumberx529.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx529.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x_values</span><span id="lstnumberx529.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx529.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.6" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx529.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx529.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx529.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx529.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx529.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">interval</span><span id="lstnumberx529.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">if</span><span id="lstnumberx529.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx529.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.21" class="ltx_text ltx_font_typewriter">*</span><span id="lstnumberx529.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.23" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx529.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx529.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.26" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx529.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.28" class="ltx_text ltx_font_typewriter">4)</span><span id="lstnumberx529.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.30" class="ltx_text ltx_font_typewriter">*</span><span id="lstnumberx529.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.32" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx529.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx529.34" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.35" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx529.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.37" class="ltx_text ltx_font_typewriter">6)</span><span id="lstnumberx529.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.39" class="ltx_text ltx_font_typewriter">*</span><span id="lstnumberx529.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.41" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx529.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx529.43" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.44" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx529.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.46" class="ltx_text ltx_font_typewriter">7)</span><span id="lstnumberx529.47" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.48" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx529.49" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.50" class="ltx_text ltx_font_typewriter">16</span><span id="lstnumberx529.51" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.52" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx529.53" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx529.54" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx529.55" class="ltx_text ltx_font_typewriter">]</span>
</div>
<div id="lstnumberx530" class="ltx_listingline">
</div>
<div id="lstnumberx531" class="ltx_listingline">
<span id="lstnumberx531.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx531.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx531.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx531.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Calculate</span><span id="lstnumberx531.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx531.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx531.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx531.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx531.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx531.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx531.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx531.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">these</span><span id="lstnumberx531.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx531.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx531.15" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx531.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">values</span>
</div>
<div id="lstnumberx532" class="ltx_listingline">
<span id="lstnumberx532.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx532.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_x</span><span id="lstnumberx532.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx532.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx532.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx532.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum</span><span id="lstnumberx532.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx532.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x_values</span><span id="lstnumberx532.9" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx533" class="ltx_listingline">
</div>
<div id="lstnumberx534" class="ltx_listingline">
<span id="lstnumberx534.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx534.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx534.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx534.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_x</span>
</div>
<div id="lstnumberx535" class="ltx_listingline">
</div>
<div id="lstnumberx536" class="ltx_listingline">
<span id="lstnumberx536.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx536.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx536.3" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx536.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx536.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sum_of_x</span><span id="lstnumberx536.6" class="ltx_text ltx_font_typewriter">()</span>
</div>
<div id="lstnumberx537" class="ltx_listingline">
<span id="lstnumberx537.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">print</span><span id="lstnumberx537.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx537.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx537.4" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx538" class="ltx_listingline">
<span id="lstnumberx538.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx539" class="ltx_listingline">
<span id="lstnumberx539.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx539.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span>
</div>
<div id="lstnumberx540" class="ltx_listingline">
<span id="lstnumberx540.1" class="ltx_text ltx_font_typewriter" style="background-color:#F47874;">SyntaxError: invalid decimal literal</span>
</div>
<div id="lstnumberx541" class="ltx_listingline">
<span id="lstnumberx541.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx542" class="ltx_listingline">
<span id="lstnumberx542.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Reach</span><span id="lstnumberx542.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx542.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">max</span><span id="lstnumberx542.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx542.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">function</span><span id="lstnumberx542.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx542.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">call</span><span id="lstnumberx542.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx542.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">limit</span><span id="lstnumberx542.10" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx543" class="ltx_listingline">
</div>
<div id="lstnumberx544" class="ltx_listingline">
<span id="lstnumberx544.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Execution</span><span id="lstnumberx544.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx544.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx544.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">None</span>
</div>
<div id="lstnumberx545" class="ltx_listingline">
<span id="lstnumberx545.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Prediction</span><span id="lstnumberx545.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx545.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx545.4" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx545.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">SyntaxError</span><span id="lstnumberx545.6" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx545.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">invaliddecimalliteral</span><span id="lstnumberx545.8" class="ltx_text ltx_font_typewriter">’</span>
</div>
<div id="lstnumberx546" class="ltx_listingline">
<span id="lstnumberx546.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Ground</span><span id="lstnumberx546.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx546.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Truth</span><span id="lstnumberx546.4" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx546.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx546.6" class="ltx_text ltx_font_typewriter">’26’</span>
</div>
<div id="lstnumberx547" class="ltx_listingline">
<span id="lstnumberx547.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Score</span><span id="lstnumberx547.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx547.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx547.4" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx547.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">False</span><span id="lstnumberx547.6" class="ltx_text ltx_font_typewriter">]</span>
</div>
<div id="lstnumberx548" class="ltx_listingline">
<span id="lstnumberx548.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Reference</span><span id="lstnumberx548.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">CoT</span><span id="lstnumberx548.4" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx548.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.6" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx548.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">We</span><span id="lstnumberx548.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">could</span><span id="lstnumberx548.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">check</span><span id="lstnumberx548.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx548.14" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx548.16" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx548.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx548.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">separately</span><span id="lstnumberx548.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx548.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">each</span><span id="lstnumberx548.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">integer</span><span id="lstnumberx548.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x$</span><span id="lstnumberx548.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx548.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$0$</span><span id="lstnumberx548.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx548.34" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$8$</span><span id="lstnumberx548.36" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx548.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx548.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">example</span><span id="lstnumberx548.41" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.42" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx548.44" class="ltx_text ltx_font_typewriter">(0)\\</span><span id="lstnumberx548.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">approx</span><span id="lstnumberx548.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.47" class="ltx_text ltx_font_typewriter">3.8</span><span id="lstnumberx548.48" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx548.49" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.50" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">so</span><span id="lstnumberx548.52" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.53" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx548.54" class="ltx_text ltx_font_typewriter">(0)&gt;0</span><span id="lstnumberx548.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx548.56" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.57" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.58" class="ltx_text ltx_lst_identifier ltx_font_typewriter">but</span><span id="lstnumberx548.59" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.60" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx548.61" class="ltx_text ltx_font_typewriter">(1)\\</span><span id="lstnumberx548.62" class="ltx_text ltx_lst_identifier ltx_font_typewriter">approx</span><span id="lstnumberx548.63" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.64" class="ltx_text ltx_font_typewriter">-0.7</span><span id="lstnumberx548.65" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx548.66" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.67" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.68" class="ltx_text ltx_lst_identifier ltx_font_typewriter">so</span><span id="lstnumberx548.69" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.70" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx548.71" class="ltx_text ltx_font_typewriter">(1)\\</span><span id="lstnumberx548.72" class="ltx_text ltx_lst_identifier ltx_font_typewriter">not</span><span id="lstnumberx548.73" class="ltx_text ltx_font_typewriter">&gt;1</span><span id="lstnumberx548.74" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx548.75" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.76" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.77" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx548.78" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.79" class="ltx_text ltx_lst_identifier ltx_font_typewriter">so</span><span id="lstnumberx548.80" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.81" class="ltx_text ltx_lst_identifier ltx_font_typewriter">on</span><span id="lstnumberx548.82" class="ltx_text ltx_font_typewriter">.\</span><span id="lstnumberx548.83" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx548.84" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx548.85" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nHowever</span><span id="lstnumberx548.86" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.87" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.88" class="ltx_text ltx_lst_identifier ltx_font_typewriter">it</span><span id="lstnumberx548.89" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.90" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx548.91" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.92" class="ltx_text ltx_lst_identifier ltx_font_typewriter">easier</span><span id="lstnumberx548.93" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.94" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx548.95" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.96" class="ltx_text ltx_lst_identifier ltx_font_typewriter">see</span><span id="lstnumberx548.97" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.98" class="ltx_text ltx_lst_identifier ltx_font_typewriter">at</span><span id="lstnumberx548.99" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.100" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx548.101" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.102" class="ltx_text ltx_lst_identifier ltx_font_typewriter">glance</span><span id="lstnumberx548.103" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.104" class="ltx_text ltx_lst_identifier ltx_font_typewriter">which</span><span id="lstnumberx548.105" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.106" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x$</span><span id="lstnumberx548.107" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.108" class="ltx_text ltx_lst_identifier ltx_font_typewriter">satisfy</span><span id="lstnumberx548.109" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.110" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx548.111" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.112" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx548.113" class="ltx_text ltx_font_typewriter">)&gt;</span><span id="lstnumberx548.114" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x$</span><span id="lstnumberx548.115" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.116" class="ltx_text ltx_lst_identifier ltx_font_typewriter">by</span><span id="lstnumberx548.117" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.118" class="ltx_text ltx_lst_identifier ltx_font_typewriter">superimposing</span><span id="lstnumberx548.119" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.120" class="ltx_text ltx_lst_identifier ltx_font_typewriter">a</span><span id="lstnumberx548.121" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.122" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx548.123" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.124" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx548.125" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.126" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx548.127" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx548.128" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x$</span><span id="lstnumberx548.129" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.130" class="ltx_text ltx_lst_identifier ltx_font_typewriter">on</span><span id="lstnumberx548.131" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.132" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx548.133" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.134" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx548.135" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.136" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx548.137" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.138" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx548.139" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx548.140" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h</span><span id="lstnumberx548.141" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.142" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx548.143" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx548.144" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx548.145" class="ltx_text ltx_font_typewriter">:\</span><span id="lstnumberx548.146" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx548.147" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx548.148" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx548.149" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx548.150" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asy</span><span id="lstnumberx548.151" class="ltx_text ltx_font_typewriter">]\</span><span id="lstnumberx548.152" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx548.153" class="ltx_text ltx_font_typewriter">((-0.75,-0.75)--(8.25,8.25),</span><span id="lstnumberx548.154" class="ltx_text ltx_lst_identifier ltx_font_typewriter">red</span><span id="lstnumberx548.155" class="ltx_text ltx_font_typewriter">+1);\</span><span id="lstnumberx548.156" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nimport</span><span id="lstnumberx548.157" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.158" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx548.159" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx548.160" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.161" class="ltx_text ltx_lst_identifier ltx_font_typewriter">size</span><span id="lstnumberx548.162" class="ltx_text ltx_font_typewriter">(8</span><span id="lstnumberx548.163" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cm</span><span id="lstnumberx548.164" class="ltx_text ltx_font_typewriter">);</span><span id="lstnumberx548.165" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.166" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx548.167" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.168" class="ltx_text ltx_lst_identifier ltx_font_typewriter">lsf</span><span id="lstnumberx548.169" class="ltx_text ltx_font_typewriter">=0.5;</span><span id="lstnumberx548.170" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.171" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pen</span><span id="lstnumberx548.172" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.173" class="ltx_text ltx_lst_identifier ltx_font_typewriter">dps</span><span id="lstnumberx548.174" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx548.175" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linewidth</span><span id="lstnumberx548.176" class="ltx_text ltx_font_typewriter">(0.7)+</span><span id="lstnumberx548.177" class="ltx_text ltx_lst_identifier ltx_font_typewriter">fontsize</span><span id="lstnumberx548.178" class="ltx_text ltx_font_typewriter">(10);</span><span id="lstnumberx548.179" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.180" class="ltx_text ltx_lst_identifier ltx_font_typewriter">defaultpen</span><span id="lstnumberx548.181" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.182" class="ltx_text ltx_lst_identifier ltx_font_typewriter">dps</span><span id="lstnumberx548.183" class="ltx_text ltx_font_typewriter">);</span><span id="lstnumberx548.184" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.185" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pen</span><span id="lstnumberx548.186" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.187" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ds</span><span id="lstnumberx548.188" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx548.189" class="ltx_text ltx_lst_identifier ltx_font_typewriter">black</span><span id="lstnumberx548.190" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx548.191" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.192" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx548.193" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.194" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx548.195" class="ltx_text ltx_font_typewriter">=-0.75,</span><span id="lstnumberx548.196" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx548.197" class="ltx_text ltx_font_typewriter">=8.25,</span><span id="lstnumberx548.198" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx548.199" class="ltx_text ltx_font_typewriter">=-1.25,</span><span id="lstnumberx548.200" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx548.201" class="ltx_text ltx_font_typewriter">=10.25;\</span><span id="lstnumberx548.202" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx548.203" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx548.204" class="ltx_text ltx_lst_identifier ltx_font_typewriter">npen</span><span id="lstnumberx548.205" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.206" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cqcqcq</span><span id="lstnumberx548.207" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx548.208" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rgb</span><span id="lstnumberx548.209" class="ltx_text ltx_font_typewriter">(0.75,0.75,0.75);\</span><span id="lstnumberx548.210" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx548.211" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx548.212" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx548.213" class="ltx_text ltx_font_typewriter">/*</span><span id="lstnumberx548.214" class="ltx_text ltx_lst_identifier ltx_font_typewriter">grid</span><span id="lstnumberx548.215" class="ltx_text ltx_font_typewriter">*/</span><span id="lstnumberx548.216" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.217" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pen</span><span id="lstnumberx548.218" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.219" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gs</span><span id="lstnumberx548.220" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx548.221" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linewidth</span><span id="lstnumberx548.222" class="ltx_text ltx_font_typewriter">(0.7)+</span><span id="lstnumberx548.223" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cqcqcq</span><span id="lstnumberx548.224" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx548.225" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linetype</span><span id="lstnumberx548.226" class="ltx_text ltx_font_typewriter">("2</span><span id="lstnumberx548.227" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.228" class="ltx_text ltx_font_typewriter">2");</span><span id="lstnumberx548.229" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.230" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx548.231" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.232" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx548.233" class="ltx_text ltx_font_typewriter">=1,</span><span id="lstnumberx548.234" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx548.235" class="ltx_text ltx_font_typewriter">=1;\</span><span id="lstnumberx548.236" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nfor</span><span id="lstnumberx548.237" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.238" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx548.239" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.240" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx548.241" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx548.242" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ceil</span><span id="lstnumberx548.243" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.244" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx548.245" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx548.246" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx548.247" class="ltx_text ltx_font_typewriter">)*</span><span id="lstnumberx548.248" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx548.249" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx548.250" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx548.251" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx548.252" class="ltx_text ltx_lst_identifier ltx_font_typewriter">floor</span><span id="lstnumberx548.253" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.254" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx548.255" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx548.256" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx548.257" class="ltx_text ltx_font_typewriter">)*</span><span id="lstnumberx548.258" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx548.259" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx548.260" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx548.261" class="ltx_text ltx_font_typewriter">+=</span><span id="lstnumberx548.262" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gx</span><span id="lstnumberx548.263" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx548.264" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.265" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx548.266" class="ltx_text ltx_font_typewriter">((</span><span id="lstnumberx548.267" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx548.268" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.269" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx548.270" class="ltx_text ltx_font_typewriter">)--(</span><span id="lstnumberx548.271" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx548.272" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.273" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx548.274" class="ltx_text ltx_font_typewriter">),</span><span id="lstnumberx548.275" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gs</span><span id="lstnumberx548.276" class="ltx_text ltx_font_typewriter">);</span><span id="lstnumberx548.277" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.278" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx548.279" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.280" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx548.281" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.282" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx548.283" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx548.284" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ceil</span><span id="lstnumberx548.285" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.286" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx548.287" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx548.288" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx548.289" class="ltx_text ltx_font_typewriter">)*</span><span id="lstnumberx548.290" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx548.291" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx548.292" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx548.293" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx548.294" class="ltx_text ltx_lst_identifier ltx_font_typewriter">floor</span><span id="lstnumberx548.295" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.296" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx548.297" class="ltx_text ltx_font_typewriter">/</span><span id="lstnumberx548.298" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx548.299" class="ltx_text ltx_font_typewriter">)*</span><span id="lstnumberx548.300" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx548.301" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx548.302" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx548.303" class="ltx_text ltx_font_typewriter">+=</span><span id="lstnumberx548.304" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gy</span><span id="lstnumberx548.305" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx548.306" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.307" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx548.308" class="ltx_text ltx_font_typewriter">((</span><span id="lstnumberx548.309" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx548.310" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.311" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx548.312" class="ltx_text ltx_font_typewriter">)--(</span><span id="lstnumberx548.313" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx548.314" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.315" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx548.316" class="ltx_text ltx_font_typewriter">),</span><span id="lstnumberx548.317" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gs</span><span id="lstnumberx548.318" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx548.319" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx548.320" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx548.321" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nLabel</span><span id="lstnumberx548.322" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.323" class="ltx_text ltx_lst_identifier ltx_font_typewriter">laxis</span><span id="lstnumberx548.324" class="ltx_text ltx_font_typewriter">;</span><span id="lstnumberx548.325" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.326" class="ltx_text ltx_lst_identifier ltx_font_typewriter">laxis</span><span id="lstnumberx548.327" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx548.328" class="ltx_text ltx_lst_identifier ltx_font_typewriter">p</span><span id="lstnumberx548.329" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx548.330" class="ltx_text ltx_lst_identifier ltx_font_typewriter">fontsize</span><span id="lstnumberx548.331" class="ltx_text ltx_font_typewriter">(10);\</span><span id="lstnumberx548.332" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx548.333" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx548.334" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nxaxis</span><span id="lstnumberx548.335" class="ltx_text ltx_font_typewriter">("",</span><span id="lstnumberx548.336" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx548.337" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.338" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx548.339" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.340" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Ticks</span><span id="lstnumberx548.341" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.342" class="ltx_text ltx_lst_identifier ltx_font_typewriter">laxis</span><span id="lstnumberx548.343" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.344" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Step</span><span id="lstnumberx548.345" class="ltx_text ltx_font_typewriter">=1.0,</span><span id="lstnumberx548.346" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Size</span><span id="lstnumberx548.347" class="ltx_text ltx_font_typewriter">=2,</span><span id="lstnumberx548.348" class="ltx_text ltx_lst_identifier ltx_font_typewriter">NoZero</span><span id="lstnumberx548.349" class="ltx_text ltx_font_typewriter">),</span><span id="lstnumberx548.350" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Arrows</span><span id="lstnumberx548.351" class="ltx_text ltx_font_typewriter">(6),</span><span id="lstnumberx548.352" class="ltx_text ltx_lst_identifier ltx_font_typewriter">above</span><span id="lstnumberx548.353" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx548.354" class="ltx_text ltx_lst_identifier ltx_font_typewriter">true</span><span id="lstnumberx548.355" class="ltx_text ltx_font_typewriter">);</span><span id="lstnumberx548.356" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.357" class="ltx_text ltx_lst_identifier ltx_font_typewriter">yaxis</span><span id="lstnumberx548.358" class="ltx_text ltx_font_typewriter">("",</span><span id="lstnumberx548.359" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx548.360" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.361" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx548.362" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.363" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Ticks</span><span id="lstnumberx548.364" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.365" class="ltx_text ltx_lst_identifier ltx_font_typewriter">laxis</span><span id="lstnumberx548.366" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.367" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Step</span><span id="lstnumberx548.368" class="ltx_text ltx_font_typewriter">=1.0,</span><span id="lstnumberx548.369" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Size</span><span id="lstnumberx548.370" class="ltx_text ltx_font_typewriter">=2,</span><span id="lstnumberx548.371" class="ltx_text ltx_lst_identifier ltx_font_typewriter">NoZero</span><span id="lstnumberx548.372" class="ltx_text ltx_font_typewriter">),</span><span id="lstnumberx548.373" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Arrows</span><span id="lstnumberx548.374" class="ltx_text ltx_font_typewriter">(6),</span><span id="lstnumberx548.375" class="ltx_text ltx_lst_identifier ltx_font_typewriter">above</span><span id="lstnumberx548.376" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx548.377" class="ltx_text ltx_lst_identifier ltx_font_typewriter">true</span><span id="lstnumberx548.378" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx548.379" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nreal</span><span id="lstnumberx548.380" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.381" class="ltx_text ltx_lst_identifier ltx_font_typewriter">f1</span><span id="lstnumberx548.382" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.383" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx548.384" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.385" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx548.386" class="ltx_text ltx_font_typewriter">){</span><span id="lstnumberx548.387" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx548.388" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.389" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.390" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx548.391" class="ltx_text ltx_font_typewriter">-0.5)*(</span><span id="lstnumberx548.392" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx548.393" class="ltx_text ltx_font_typewriter">-2.5)*(</span><span id="lstnumberx548.394" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx548.395" class="ltx_text ltx_font_typewriter">-6.5)*(</span><span id="lstnumberx548.396" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx548.397" class="ltx_text ltx_font_typewriter">-7.5)/16+</span><span id="lstnumberx548.398" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx548.399" class="ltx_text ltx_font_typewriter">;}\</span><span id="lstnumberx548.400" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx548.401" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.402" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx548.403" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.404" class="ltx_text ltx_lst_identifier ltx_font_typewriter">f1</span><span id="lstnumberx548.405" class="ltx_text ltx_font_typewriter">,-0.25,8.25),</span><span id="lstnumberx548.406" class="ltx_text ltx_lst_identifier ltx_font_typewriter">linewidth</span><span id="lstnumberx548.407" class="ltx_text ltx_font_typewriter">(1));\</span><span id="lstnumberx548.408" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nclip</span><span id="lstnumberx548.409" class="ltx_text ltx_font_typewriter">((</span><span id="lstnumberx548.410" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx548.411" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.412" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx548.413" class="ltx_text ltx_font_typewriter">)--(</span><span id="lstnumberx548.414" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmin</span><span id="lstnumberx548.415" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.416" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx548.417" class="ltx_text ltx_font_typewriter">)--(</span><span id="lstnumberx548.418" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx548.419" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.420" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymax</span><span id="lstnumberx548.421" class="ltx_text ltx_font_typewriter">)--(</span><span id="lstnumberx548.422" class="ltx_text ltx_lst_identifier ltx_font_typewriter">xmax</span><span id="lstnumberx548.423" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.424" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ymin</span><span id="lstnumberx548.425" class="ltx_text ltx_font_typewriter">)--</span><span id="lstnumberx548.426" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cycle</span><span id="lstnumberx548.427" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx548.428" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nlabel</span><span id="lstnumberx548.429" class="ltx_text ltx_font_typewriter">("</span><span id="lstnumberx548.430" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx548.431" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx548.432" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h</span><span id="lstnumberx548.433" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.434" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx548.435" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx548.436" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx548.437" class="ltx_text ltx_font_typewriter">",(8.5,8),</span><span id="lstnumberx548.438" class="ltx_text ltx_lst_identifier ltx_font_typewriter">E</span><span id="lstnumberx548.439" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx548.440" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndot</span><span id="lstnumberx548.441" class="ltx_text ltx_font_typewriter">((0,0),</span><span id="lstnumberx548.442" class="ltx_text ltx_lst_identifier ltx_font_typewriter">blue</span><span id="lstnumberx548.443" class="ltx_text ltx_font_typewriter">);</span><span id="lstnumberx548.444" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.445" class="ltx_text ltx_lst_identifier ltx_font_typewriter">dot</span><span id="lstnumberx548.446" class="ltx_text ltx_font_typewriter">((3,3),</span><span id="lstnumberx548.447" class="ltx_text ltx_lst_identifier ltx_font_typewriter">blue</span><span id="lstnumberx548.448" class="ltx_text ltx_font_typewriter">);</span><span id="lstnumberx548.449" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.450" class="ltx_text ltx_lst_identifier ltx_font_typewriter">dot</span><span id="lstnumberx548.451" class="ltx_text ltx_font_typewriter">((4,4),</span><span id="lstnumberx548.452" class="ltx_text ltx_lst_identifier ltx_font_typewriter">blue</span><span id="lstnumberx548.453" class="ltx_text ltx_font_typewriter">);</span><span id="lstnumberx548.454" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.455" class="ltx_text ltx_lst_identifier ltx_font_typewriter">dot</span><span id="lstnumberx548.456" class="ltx_text ltx_font_typewriter">((5,5),</span><span id="lstnumberx548.457" class="ltx_text ltx_lst_identifier ltx_font_typewriter">blue</span><span id="lstnumberx548.458" class="ltx_text ltx_font_typewriter">);</span><span id="lstnumberx548.459" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.460" class="ltx_text ltx_lst_identifier ltx_font_typewriter">dot</span><span id="lstnumberx548.461" class="ltx_text ltx_font_typewriter">((6,6),</span><span id="lstnumberx548.462" class="ltx_text ltx_lst_identifier ltx_font_typewriter">blue</span><span id="lstnumberx548.463" class="ltx_text ltx_font_typewriter">);</span><span id="lstnumberx548.464" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.465" class="ltx_text ltx_lst_identifier ltx_font_typewriter">dot</span><span id="lstnumberx548.466" class="ltx_text ltx_font_typewriter">((8,8),</span><span id="lstnumberx548.467" class="ltx_text ltx_lst_identifier ltx_font_typewriter">blue</span><span id="lstnumberx548.468" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx548.469" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx548.470" class="ltx_text ltx_font_typewriter">[/</span><span id="lstnumberx548.471" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asy</span><span id="lstnumberx548.472" class="ltx_text ltx_font_typewriter">]\</span><span id="lstnumberx548.473" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx548.474" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx548.475" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nThe</span><span id="lstnumberx548.476" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.477" class="ltx_text ltx_lst_identifier ltx_font_typewriter">six</span><span id="lstnumberx548.478" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.479" class="ltx_text ltx_lst_identifier ltx_font_typewriter">blue</span><span id="lstnumberx548.480" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.481" class="ltx_text ltx_lst_identifier ltx_font_typewriter">dots</span><span id="lstnumberx548.482" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.483" class="ltx_text ltx_lst_identifier ltx_font_typewriter">above</span><span id="lstnumberx548.484" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.485" class="ltx_text ltx_lst_identifier ltx_font_typewriter">mark</span><span id="lstnumberx548.486" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.487" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx548.488" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.489" class="ltx_text ltx_lst_identifier ltx_font_typewriter">integer</span><span id="lstnumberx548.490" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.491" class="ltx_text ltx_lst_identifier ltx_font_typewriter">points</span><span id="lstnumberx548.492" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.493" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx548.494" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.495" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx548.496" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.497" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx548.498" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx548.499" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx548.500" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.501" class="ltx_text ltx_lst_identifier ltx_font_typewriter">which</span><span id="lstnumberx548.502" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.503" class="ltx_text ltx_lst_identifier ltx_font_typewriter">lie</span><span id="lstnumberx548.504" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.505" class="ltx_text ltx_lst_identifier ltx_font_typewriter">below</span><span id="lstnumberx548.506" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.507" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx548.508" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.509" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx548.510" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.511" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx548.512" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.513" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx548.514" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx548.515" class="ltx_text ltx_lst_identifier ltx_font_typewriter">h</span><span id="lstnumberx548.516" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.517" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx548.518" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx548.519" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx548.520" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx548.521" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.522" class="ltx_text ltx_lst_identifier ltx_font_typewriter">indicating</span><span id="lstnumberx548.523" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.524" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx548.525" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.526" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$h</span><span id="lstnumberx548.527" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx548.528" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx548.529" class="ltx_text ltx_font_typewriter">)&gt;</span><span id="lstnumberx548.530" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x$</span><span id="lstnumberx548.531" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx548.532" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.533" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Their</span><span id="lstnumberx548.534" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.535" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x$</span><span id="lstnumberx548.536" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx548.537" class="ltx_text ltx_lst_identifier ltx_font_typewriter">coordinates</span><span id="lstnumberx548.538" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.539" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx548.540" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.541" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$0</span><span id="lstnumberx548.542" class="ltx_text ltx_font_typewriter">,3,4,5,6,8,</span><span id="lstnumberx548.543" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx548.544" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.545" class="ltx_text ltx_lst_identifier ltx_font_typewriter">which</span><span id="lstnumberx548.546" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.547" class="ltx_text ltx_lst_identifier ltx_font_typewriter">add</span><span id="lstnumberx548.548" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.549" class="ltx_text ltx_lst_identifier ltx_font_typewriter">up</span><span id="lstnumberx548.550" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.551" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx548.552" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx548.553" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx548.554" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx548.555" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx548.556" class="ltx_text ltx_font_typewriter">{26}</span><span id="lstnumberx548.557" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx548.558" class="ltx_text ltx_font_typewriter">.’</span>
</div>
</div>
</figure>
<figure id="LST10" class="ltx_float ltx_lstlisting">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float">Listing 10: </span>Failure case for <span id="LST10.4.1" class="ltx_text ltx_font_smallcaps">ToRA</span>: <span id="LST10.5.2" class="ltx_text ltx_font_bold">Runtime Error</span>. <span id="LST10.6.3" class="ltx_text ltx_font_smallcaps">ToRA</span> failed to fall back from program timeout to a rationale-based solution. For Intermediate Algebra and Precalculus problems like this example, direct brute-force solutions are often infeasible, resulting in exceptions like timeout.
Addressing these challenges requires complex reasoning over math expressions, and may involve sophisticated problem-solving and proof techniques like forward and backward reasoning <cite class="ltx_cite ltx_citemacro_citep">(Jiang et al., <a href="#bib.bib17" title="" class="ltx_ref">2023</a>)</cite>, as well as result verification <cite class="ltx_cite ltx_citemacro_citep">(Gou et al., <a href="#bib.bib11" title="" class="ltx_ref">2023</a>; Zhou et al., <a href="#bib.bib51" title="" class="ltx_ref">2023a</a>)</cite>.</figcaption>
<div id="LST10.7" class="ltx_listing ltx_lstlisting ltx_framed ltx_framed_topbottom ltx_listing">
<div class="ltx_listing_data"><a href="data:text/plain;base64,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" download="section/cases/failure/6_runtime_error.md">⬇</a></div>
<div id="lstnumberx549" class="ltx_listingline">
<span id="lstnumberx549.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Idx</span><span id="lstnumberx549.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx549.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx549.4" class="ltx_text ltx_font_typewriter">2261</span>
</div>
<div id="lstnumberx550" class="ltx_listingline">
<span id="lstnumberx550.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Yype</span><span id="lstnumberx550.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx550.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx550.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Intermediate</span><span id="lstnumberx550.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx550.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Algebra</span>
</div>
<div id="lstnumberx551" class="ltx_listingline">
<span id="lstnumberx551.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Level</span><span id="lstnumberx551.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx551.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx551.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Level</span><span id="lstnumberx551.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx551.6" class="ltx_text ltx_font_typewriter">3</span>
</div>
<div id="lstnumberx552" class="ltx_listingline">
<span id="lstnumberx552.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Question</span><span id="lstnumberx552.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx552.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.4" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx552.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">For</span><span id="lstnumberx552.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">some</span><span id="lstnumberx552.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">positive</span><span id="lstnumberx552.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">integers</span><span id="lstnumberx552.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$m$</span><span id="lstnumberx552.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx552.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$n</span><span id="lstnumberx552.18" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx552.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx552.20" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx552.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx552.22" class="ltx_text ltx_font_typewriter">\\[\\</span><span id="lstnumberx552.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx552.24" class="ltx_text ltx_font_typewriter">{23</span><span id="lstnumberx552.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.26" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx552.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.28" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx552.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx552.30" class="ltx_text ltx_font_typewriter">{28}}</span><span id="lstnumberx552.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.32" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx552.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.34" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx552.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx552.36" class="ltx_text ltx_font_typewriter">{23</span><span id="lstnumberx552.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.38" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx552.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.40" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx552.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx552.42" class="ltx_text ltx_font_typewriter">{28}}</span><span id="lstnumberx552.43" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.44" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx552.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.46" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx552.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx552.48" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx552.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">m</span><span id="lstnumberx552.50" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.51" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx552.52" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.53" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx552.54" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx552.55" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx552.56" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx552.57" class="ltx_text ltx_font_typewriter">}}.\\]</span><span id="lstnumberx552.58" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Enter</span><span id="lstnumberx552.59" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.60" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx552.61" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.62" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ordered</span><span id="lstnumberx552.63" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.64" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pair</span><span id="lstnumberx552.65" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx552.66" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx552.67" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx552.68" class="ltx_text ltx_lst_identifier ltx_font_typewriter">m</span><span id="lstnumberx552.69" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx552.70" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx552.71" class="ltx_text ltx_font_typewriter">).</span><span id="lstnumberx552.72" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx552.73" class="ltx_text ltx_font_typewriter">’</span>
</div>
<div id="lstnumberx553" class="ltx_listingline">
<span id="lstnumberx553.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solution</span><span id="lstnumberx553.2" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx554" class="ltx_listingline">
<span id="lstnumberx554.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx554.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span>
</div>
<div id="lstnumberx555" class="ltx_listingline">
<span id="lstnumberx555.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx555.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx555.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx555.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx555.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx555.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx555.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx555.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx555.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx555.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve</span><span id="lstnumberx555.11" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx555.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx555.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span>
</div>
<div id="lstnumberx556" class="ltx_listingline">
</div>
<div id="lstnumberx557" class="ltx_listingline">
<span id="lstnumberx557.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">def</span><span id="lstnumberx557.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx557.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">find_m_n</span><span id="lstnumberx557.4" class="ltx_text ltx_font_typewriter">():</span>
</div>
<div id="lstnumberx558" class="ltx_listingline">
<span id="lstnumberx558.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx558.2" class="ltx_text ltx_font_typewriter">"""</span><span id="lstnumberx558.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">For</span><span id="lstnumberx558.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx558.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">some</span><span id="lstnumberx558.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx558.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">positive</span><span id="lstnumberx558.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx558.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">integers</span><span id="lstnumberx558.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx558.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$m$</span><span id="lstnumberx558.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx558.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx558.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx558.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$n</span><span id="lstnumberx558.16" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx558.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span>
</div>
<div id="lstnumberx559" class="ltx_listingline">
<span id="lstnumberx559.1" class="ltx_text ltx_font_typewriter">\[\</span><span id="lstnumberx559.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx559.3" class="ltx_text ltx_font_typewriter">{23</span><span id="lstnumberx559.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx559.5" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx559.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx559.7" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx559.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx559.9" class="ltx_text ltx_font_typewriter">{28}}</span><span id="lstnumberx559.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx559.11" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx559.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx559.13" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx559.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx559.15" class="ltx_text ltx_font_typewriter">{23</span><span id="lstnumberx559.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx559.17" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx559.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx559.19" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx559.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx559.21" class="ltx_text ltx_font_typewriter">{28}}</span><span id="lstnumberx559.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx559.23" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx559.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx559.25" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx559.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx559.27" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx559.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">m</span><span id="lstnumberx559.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx559.30" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx559.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx559.32" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx559.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx559.34" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx559.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx559.36" class="ltx_text ltx_font_typewriter">}}.\]</span><span id="lstnumberx559.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Enter</span><span id="lstnumberx559.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx559.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx559.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx559.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ordered</span><span id="lstnumberx559.42" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx559.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pair</span><span id="lstnumberx559.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx559.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx559.46" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx559.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">m</span><span id="lstnumberx559.48" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx559.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx559.50" class="ltx_text ltx_font_typewriter">).</span><span id="lstnumberx559.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx559.52" class="ltx_text ltx_font_typewriter">"""</span>
</div>
<div id="lstnumberx560" class="ltx_listingline">
<span id="lstnumberx560.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx560.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">m</span><span id="lstnumberx560.3" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx560.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx560.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx560.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx560.7" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx560.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx560.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span><span id="lstnumberx560.10" class="ltx_text ltx_font_typewriter">(’</span><span id="lstnumberx560.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">m</span><span id="lstnumberx560.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx560.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx560.14" class="ltx_text ltx_font_typewriter">’)</span>
</div>
<div id="lstnumberx561" class="ltx_listingline">
</div>
<div id="lstnumberx562" class="ltx_listingline">
<span id="lstnumberx562.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx562.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx562.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx562.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Define</span><span id="lstnumberx562.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx562.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx562.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx562.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">given</span><span id="lstnumberx562.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx562.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equation</span>
</div>
<div id="lstnumberx563" class="ltx_listingline">
<span id="lstnumberx563.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx563.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equation</span><span id="lstnumberx563.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx563.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx563.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx563.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx563.7" class="ltx_text ltx_font_typewriter">(23</span><span id="lstnumberx563.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx563.9" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx563.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx563.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx563.12" class="ltx_text ltx_font_typewriter">(28))</span><span id="lstnumberx563.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx563.14" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx563.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx563.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx563.17" class="ltx_text ltx_font_typewriter">(23</span><span id="lstnumberx563.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx563.19" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx563.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx563.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx563.22" class="ltx_text ltx_font_typewriter">(28))</span><span id="lstnumberx563.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx563.24" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx563.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx563.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx563.27" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx563.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">m</span><span id="lstnumberx563.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx563.30" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx563.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx563.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx563.33" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx563.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx563.35" class="ltx_text ltx_font_typewriter">))</span>
</div>
<div id="lstnumberx564" class="ltx_listingline">
</div>
<div id="lstnumberx565" class="ltx_listingline">
<span id="lstnumberx565.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx565.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx565.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx565.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solve</span><span id="lstnumberx565.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx565.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx565.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx565.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equation</span><span id="lstnumberx565.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx565.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx565.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx565.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">m</span><span id="lstnumberx565.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx565.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx565.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx565.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span>
</div>
<div id="lstnumberx566" class="ltx_listingline">
<span id="lstnumberx566.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx566.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solutions</span><span id="lstnumberx566.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx566.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx566.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx566.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve</span><span id="lstnumberx566.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx566.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equation</span><span id="lstnumberx566.9" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx566.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx566.11" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx566.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">m</span><span id="lstnumberx566.13" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx566.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx566.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx566.16" class="ltx_text ltx_font_typewriter">))</span>
</div>
<div id="lstnumberx567" class="ltx_listingline">
</div>
<div id="lstnumberx568" class="ltx_listingline">
<span id="lstnumberx568.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx568.2" class="ltx_text ltx_font_typewriter">#</span><span id="lstnumberx568.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx568.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Filter</span><span id="lstnumberx568.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx568.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">out</span><span id="lstnumberx568.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx568.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx568.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx568.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solutions</span><span id="lstnumberx568.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx568.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">where</span><span id="lstnumberx568.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx568.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">m</span><span id="lstnumberx568.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx568.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx568.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx568.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx568.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx568.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx568.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx568.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">not</span><span id="lstnumberx568.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx568.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">positive</span><span id="lstnumberx568.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx568.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">integers</span>
</div>
<div id="lstnumberx569" class="ltx_listingline">
<span id="lstnumberx569.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx569.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solutions</span><span id="lstnumberx569.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx569.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.6" class="ltx_text ltx_font_typewriter">[(</span><span id="lstnumberx569.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sol_m</span><span id="lstnumberx569.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx569.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sol_n</span><span id="lstnumberx569.11" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx569.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx569.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sol_m</span><span id="lstnumberx569.16" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx569.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sol_n</span><span id="lstnumberx569.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx569.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solutions</span><span id="lstnumberx569.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">if</span><span id="lstnumberx569.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sol_m</span><span id="lstnumberx569.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.28" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx569.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.30" class="ltx_text ltx_font_typewriter">0</span><span id="lstnumberx569.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx569.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sol_n</span><span id="lstnumberx569.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.36" class="ltx_text ltx_font_typewriter">&gt;</span><span id="lstnumberx569.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.38" class="ltx_text ltx_font_typewriter">0</span><span id="lstnumberx569.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx569.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sol_m</span><span id="lstnumberx569.43" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx569.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is_integer</span><span id="lstnumberx569.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx569.47" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx569.48" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sol_n</span><span id="lstnumberx569.49" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx569.50" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is_integer</span><span id="lstnumberx569.51" class="ltx_text ltx_font_typewriter">]</span>
</div>
<div id="lstnumberx570" class="ltx_listingline">
</div>
<div id="lstnumberx571" class="ltx_listingline">
<span id="lstnumberx571.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx571.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx571.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx571.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solutions</span><span id="lstnumberx571.5" class="ltx_text ltx_font_typewriter">[0]</span>
</div>
<div id="lstnumberx572" class="ltx_listingline">
</div>
<div id="lstnumberx573" class="ltx_listingline">
<span id="lstnumberx573.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx573.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx573.3" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx573.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx573.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">find_m_n</span><span id="lstnumberx573.6" class="ltx_text ltx_font_typewriter">()</span>
</div>
<div id="lstnumberx574" class="ltx_listingline">
<span id="lstnumberx574.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">print</span><span id="lstnumberx574.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx574.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx574.4" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx575" class="ltx_listingline">
<span id="lstnumberx575.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx576" class="ltx_listingline">
<span id="lstnumberx576.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx576.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span>
</div>
<div id="lstnumberx577" class="ltx_listingline">
<span id="lstnumberx577.1" class="ltx_text ltx_font_typewriter" style="background-color:#F47874;">TypeError: cannot determine truth value of Relational</span>
</div>
<div id="lstnumberx578" class="ltx_listingline">
<span id="lstnumberx578.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx579" class="ltx_listingline">
<span id="lstnumberx579.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx579.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">error</span><span id="lstnumberx579.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx579.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx579.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">code</span><span id="lstnumberx579.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">above</span><span id="lstnumberx579.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx579.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">due</span><span id="lstnumberx579.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx579.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx579.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">fact</span><span id="lstnumberx579.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">that</span><span id="lstnumberx579.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx579.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.27" class="ltx_text ltx_font_typewriter">‘</span><span id="lstnumberx579.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve</span><span id="lstnumberx579.29" class="ltx_text ltx_font_typewriter">‘</span><span id="lstnumberx579.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">function</span><span id="lstnumberx579.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx579.34" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">SymPy</span><span id="lstnumberx579.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">does</span><span id="lstnumberx579.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">not</span><span id="lstnumberx579.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">handle</span><span id="lstnumberx579.42" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">square</span><span id="lstnumberx579.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots</span><span id="lstnumberx579.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">very</span><span id="lstnumberx579.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">well</span><span id="lstnumberx579.50" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx579.51" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Instead</span><span id="lstnumberx579.53" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx579.54" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.55" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx579.56" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">can</span><span id="lstnumberx579.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">manually</span><span id="lstnumberx579.60" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve</span><span id="lstnumberx579.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx579.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.65" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equation</span><span id="lstnumberx579.66" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx579.68" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.69" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$m$</span><span id="lstnumberx579.70" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.71" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx579.72" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.73" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$n$</span><span id="lstnumberx579.74" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.75" class="ltx_text ltx_lst_identifier ltx_font_typewriter">using</span><span id="lstnumberx579.76" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.77" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx579.78" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.79" class="ltx_text ltx_lst_identifier ltx_font_typewriter">properties</span><span id="lstnumberx579.80" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.81" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx579.82" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.83" class="ltx_text ltx_lst_identifier ltx_font_typewriter">square</span><span id="lstnumberx579.84" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx579.85" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots</span><span id="lstnumberx579.86" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx580" class="ltx_listingline">
</div>
<div id="lstnumberx581" class="ltx_listingline">
<span id="lstnumberx581.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Let</span><span id="lstnumberx581.2" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx581.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">s</span><span id="lstnumberx581.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">denote</span><span id="lstnumberx581.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx581.8" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx581.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx581.10" class="ltx_text ltx_font_typewriter">{23</span><span id="lstnumberx581.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.12" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx581.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.14" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx581.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx581.16" class="ltx_text ltx_font_typewriter">{28}}</span><span id="lstnumberx581.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx581.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">as</span><span id="lstnumberx581.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$a$</span><span id="lstnumberx581.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx581.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx581.26" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx581.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx581.28" class="ltx_text ltx_font_typewriter">{23</span><span id="lstnumberx581.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.30" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx581.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.32" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx581.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx581.34" class="ltx_text ltx_font_typewriter">{28}}</span><span id="lstnumberx581.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx581.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">as</span><span id="lstnumberx581.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$b$</span><span id="lstnumberx581.40" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx581.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Then</span><span id="lstnumberx581.43" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx581.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">given</span><span id="lstnumberx581.47" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.48" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equation</span><span id="lstnumberx581.49" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.50" class="ltx_text ltx_lst_identifier ltx_font_typewriter">can</span><span id="lstnumberx581.51" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">be</span><span id="lstnumberx581.53" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.54" class="ltx_text ltx_lst_identifier ltx_font_typewriter">rewritten</span><span id="lstnumberx581.55" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.56" class="ltx_text ltx_lst_identifier ltx_font_typewriter">as</span><span id="lstnumberx581.57" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.58" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$a</span><span id="lstnumberx581.59" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.60" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx581.61" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.62" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b</span><span id="lstnumberx581.63" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.64" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx581.65" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.66" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx581.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx581.68" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx581.69" class="ltx_text ltx_lst_identifier ltx_font_typewriter">m</span><span id="lstnumberx581.70" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.71" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx581.72" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx581.73" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx581.74" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx581.75" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx581.76" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx581.77" class="ltx_text ltx_font_typewriter">}}</span><span id="lstnumberx581.78" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx581.79" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx582" class="ltx_listingline">
</div>
<div id="lstnumberx583" class="ltx_listingline">
<span id="lstnumberx583.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Squaring</span><span id="lstnumberx583.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx583.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">both</span><span id="lstnumberx583.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx583.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sides</span><span id="lstnumberx583.6" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx583.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx583.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx583.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx583.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">get</span><span id="lstnumberx583.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx583.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$a</span><span id="lstnumberx583.13" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx583.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx583.15" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx583.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx583.17" class="ltx_text ltx_font_typewriter">2</span><span id="lstnumberx583.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ab</span><span id="lstnumberx583.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx583.20" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx583.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx583.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">b</span><span id="lstnumberx583.23" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx583.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx583.25" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx583.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx583.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">m</span><span id="lstnumberx583.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx583.29" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx583.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx583.31" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx583.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx583.33" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx583.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx583.35" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx583.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx583.37" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx584" class="ltx_listingline">
</div>
<div id="lstnumberx585" class="ltx_listingline">
<span id="lstnumberx585.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Since</span><span id="lstnumberx585.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$a</span><span id="lstnumberx585.4" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx585.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.6" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx585.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.8" class="ltx_text ltx_font_typewriter">23</span><span id="lstnumberx585.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.10" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx585.11" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.12" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx585.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx585.14" class="ltx_text ltx_font_typewriter">{28}</span><span id="lstnumberx585.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx585.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx585.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$b</span><span id="lstnumberx585.20" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx585.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.22" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx585.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.24" class="ltx_text ltx_font_typewriter">23</span><span id="lstnumberx585.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.26" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx585.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.28" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx585.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx585.30" class="ltx_text ltx_font_typewriter">{28}</span><span id="lstnumberx585.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx585.32" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx585.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">we</span><span id="lstnumberx585.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">can</span><span id="lstnumberx585.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">substitute</span><span id="lstnumberx585.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">these</span><span id="lstnumberx585.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">values</span><span id="lstnumberx585.43" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">into</span><span id="lstnumberx585.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx585.47" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.48" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equation</span><span id="lstnumberx585.49" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.50" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx585.51" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">get</span><span id="lstnumberx585.53" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.54" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$28</span><span id="lstnumberx585.55" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.56" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx585.57" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.58" class="ltx_text ltx_font_typewriter">2\</span><span id="lstnumberx585.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx585.60" class="ltx_text ltx_font_typewriter">{28}</span><span id="lstnumberx585.61" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.62" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx585.63" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.64" class="ltx_text ltx_font_typewriter">23</span><span id="lstnumberx585.65" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.66" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx585.67" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.68" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx585.69" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx585.70" class="ltx_text ltx_font_typewriter">{28}</span><span id="lstnumberx585.71" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.72" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx585.73" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.74" class="ltx_text ltx_lst_identifier ltx_font_typewriter">m</span><span id="lstnumberx585.75" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.76" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx585.77" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx585.78" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx585.79" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx585.80" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx585.81" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx585.82" class="ltx_text ltx_font_typewriter">}</span><span id="lstnumberx585.83" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx585.84" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx586" class="ltx_listingline">
</div>
<div id="lstnumberx587" class="ltx_listingline">
<span id="lstnumberx587.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solving</span><span id="lstnumberx587.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx587.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">this</span><span id="lstnumberx587.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx587.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equation</span><span id="lstnumberx587.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx587.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx587.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx587.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$m$</span><span id="lstnumberx587.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx587.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx587.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx587.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$n$</span><span id="lstnumberx587.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx587.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gives</span><span id="lstnumberx587.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx587.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">us</span><span id="lstnumberx587.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx587.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$m</span><span id="lstnumberx587.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx587.21" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx587.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx587.23" class="ltx_text ltx_font_typewriter">49</span><span id="lstnumberx587.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx587.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx587.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx587.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx587.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$n</span><span id="lstnumberx587.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx587.30" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx587.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx587.32" class="ltx_text ltx_font_typewriter">72</span><span id="lstnumberx587.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx587.34" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx588" class="ltx_listingline">
</div>
<div id="lstnumberx589" class="ltx_listingline">
<span id="lstnumberx589.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Therefore</span><span id="lstnumberx589.2" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx589.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx589.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx589.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx589.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ordered</span><span id="lstnumberx589.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx589.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pair</span><span id="lstnumberx589.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx589.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx589.11" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx589.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">m</span><span id="lstnumberx589.13" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx589.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx589.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx589.16" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx589.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx589.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx589.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx589.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx589.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx589.22" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx589.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx589.24" class="ltx_text ltx_font_typewriter">{(49,</span><span id="lstnumberx589.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx589.26" class="ltx_text ltx_font_typewriter">72)}</span><span id="lstnumberx589.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx589.28" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx590" class="ltx_listingline">
</div>
<div id="lstnumberx591" class="ltx_listingline">
<span id="lstnumberx591.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Execution</span><span id="lstnumberx591.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx591.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx591.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">None</span>
</div>
<div id="lstnumberx592" class="ltx_listingline">
<span id="lstnumberx592.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Prediction</span><span id="lstnumberx592.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx592.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx592.4" class="ltx_text ltx_font_typewriter">(49,72)</span>
</div>
<div id="lstnumberx593" class="ltx_listingline">
<span id="lstnumberx593.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Ground</span><span id="lstnumberx593.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx593.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Truth</span><span id="lstnumberx593.4" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx593.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx593.6" class="ltx_text ltx_font_typewriter">(46,2004)</span>
</div>
<div id="lstnumberx594" class="ltx_listingline">
<span id="lstnumberx594.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Score</span><span id="lstnumberx594.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx594.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx594.4" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx594.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">False</span><span id="lstnumberx594.6" class="ltx_text ltx_font_typewriter">]</span>
</div>
<div id="lstnumberx595" class="ltx_listingline">
<span id="lstnumberx595.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Reference</span><span id="lstnumberx595.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">CoT</span><span id="lstnumberx595.4" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx595.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.6" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx595.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Let</span><span id="lstnumberx595.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x</span><span id="lstnumberx595.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.11" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx595.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.13" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx595.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx595.15" class="ltx_text ltx_font_typewriter">{23</span><span id="lstnumberx595.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.17" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx595.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.19" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx595.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx595.21" class="ltx_text ltx_font_typewriter">{28}}</span><span id="lstnumberx595.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.23" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx595.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.25" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx595.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx595.27" class="ltx_text ltx_font_typewriter">{23</span><span id="lstnumberx595.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.29" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx595.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.31" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx595.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx595.33" class="ltx_text ltx_font_typewriter">{28}}.</span><span id="lstnumberx595.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx595.35" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx595.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Then</span><span id="lstnumberx595.37" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx595.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx595.39" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx595.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">begin</span><span id="lstnumberx595.41" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx595.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">align</span><span id="lstnumberx595.43" class="ltx_text ltx_font_typewriter">*}\</span><span id="lstnumberx595.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nx</span><span id="lstnumberx595.45" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx595.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.47" class="ltx_text ltx_font_typewriter">&amp;=</span><span id="lstnumberx595.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.49" class="ltx_text ltx_font_typewriter">23</span><span id="lstnumberx595.50" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.51" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx595.52" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.53" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx595.54" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx595.55" class="ltx_text ltx_font_typewriter">{28}</span><span id="lstnumberx595.56" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.57" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx595.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.59" class="ltx_text ltx_font_typewriter">2</span><span id="lstnumberx595.60" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.61" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx595.62" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx595.63" class="ltx_text ltx_font_typewriter">{23</span><span id="lstnumberx595.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.65" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx595.66" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.67" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx595.68" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx595.69" class="ltx_text ltx_font_typewriter">{28}}</span><span id="lstnumberx595.70" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.71" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx595.72" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx595.73" class="ltx_text ltx_font_typewriter">{23</span><span id="lstnumberx595.74" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.75" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx595.76" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.77" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx595.78" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx595.79" class="ltx_text ltx_font_typewriter">{28}}</span><span id="lstnumberx595.80" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.81" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx595.82" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.83" class="ltx_text ltx_font_typewriter">23</span><span id="lstnumberx595.84" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.85" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx595.86" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.87" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx595.88" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx595.89" class="ltx_text ltx_font_typewriter">{28}</span><span id="lstnumberx595.90" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.91" class="ltx_text ltx_font_typewriter">\\\\\</span><span id="lstnumberx595.92" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx595.93" class="ltx_text ltx_font_typewriter">&amp;=</span><span id="lstnumberx595.94" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.95" class="ltx_text ltx_font_typewriter">46</span><span id="lstnumberx595.96" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.97" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx595.98" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.99" class="ltx_text ltx_font_typewriter">2</span><span id="lstnumberx595.100" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.101" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx595.102" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx595.103" class="ltx_text ltx_font_typewriter">{23^2</span><span id="lstnumberx595.104" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.105" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx595.106" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.107" class="ltx_text ltx_font_typewriter">28}</span><span id="lstnumberx595.108" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.109" class="ltx_text ltx_font_typewriter">\\\\\</span><span id="lstnumberx595.110" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx595.111" class="ltx_text ltx_font_typewriter">&amp;=</span><span id="lstnumberx595.112" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.113" class="ltx_text ltx_font_typewriter">46</span><span id="lstnumberx595.114" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.115" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx595.116" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.117" class="ltx_text ltx_font_typewriter">2</span><span id="lstnumberx595.118" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.119" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx595.120" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx595.121" class="ltx_text ltx_font_typewriter">{501}</span><span id="lstnumberx595.122" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.123" class="ltx_text ltx_font_typewriter">\\\\\</span><span id="lstnumberx595.124" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx595.125" class="ltx_text ltx_font_typewriter">&amp;=</span><span id="lstnumberx595.126" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.127" class="ltx_text ltx_font_typewriter">46</span><span id="lstnumberx595.128" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.129" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx595.130" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.131" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx595.132" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx595.133" class="ltx_text ltx_font_typewriter">{2004}.\</span><span id="lstnumberx595.134" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx595.135" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx595.136" class="ltx_text ltx_lst_identifier ltx_font_typewriter">end</span><span id="lstnumberx595.137" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx595.138" class="ltx_text ltx_lst_identifier ltx_font_typewriter">align</span><span id="lstnumberx595.139" class="ltx_text ltx_font_typewriter">*}</span><span id="lstnumberx595.140" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Hence</span><span id="lstnumberx595.141" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx595.142" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.143" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx595.144" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx595.145" class="ltx_text ltx_lst_identifier ltx_font_typewriter">m</span><span id="lstnumberx595.146" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx595.147" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx595.148" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx595.149" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.150" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx595.151" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx595.152" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx595.153" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx595.154" class="ltx_text ltx_font_typewriter">{(46,2004)}.</span><span id="lstnumberx595.155" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx595.156" class="ltx_text ltx_font_typewriter">’</span>
</div>
</div>
</figure>
<figure id="LST11" class="ltx_float ltx_lstlisting">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float">Listing 11: </span>Failure case for <span id="LST11.3.1" class="ltx_text ltx_font_smallcaps">ToRA</span>: <span id="LST11.4.2" class="ltx_text ltx_font_bold">Rationale-only Error</span>.</figcaption>
<div id="LST11.5" class="ltx_listing ltx_lstlisting ltx_framed ltx_framed_topbottom ltx_listing">
<div class="ltx_listing_data"><a href="data:text/plain;base64,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" download="section/cases/failure/7_rationale_only_error.md">⬇</a></div>
<div id="lstnumberx596" class="ltx_listingline">
<span id="lstnumberx596.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Idx</span><span id="lstnumberx596.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx596.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx596.4" class="ltx_text ltx_font_typewriter">2638</span>
</div>
<div id="lstnumberx597" class="ltx_listingline">
<span id="lstnumberx597.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Type</span><span id="lstnumberx597.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx597.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx597.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Intermediate</span><span id="lstnumberx597.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx597.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Algebra</span>
</div>
<div id="lstnumberx598" class="ltx_listingline">
<span id="lstnumberx598.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Level</span><span id="lstnumberx598.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx598.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx598.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Level</span><span id="lstnumberx598.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx598.6" class="ltx_text ltx_font_typewriter">1</span>
</div>
<div id="lstnumberx599" class="ltx_listingline">
<span id="lstnumberx599.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Question</span><span id="lstnumberx599.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx599.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.4" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx599.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx599.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx599.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx599.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx599.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.13" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx599.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">f</span><span id="lstnumberx599.16" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.18" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx599.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx599.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx599.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">shown</span><span id="lstnumberx599.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">below</span><span id="lstnumberx599.26" class="ltx_text ltx_font_typewriter">.\</span><span id="lstnumberx599.27" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.28" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.30" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx599.31" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asy</span><span id="lstnumberx599.32" class="ltx_text ltx_font_typewriter">]\</span><span id="lstnumberx599.33" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nunitsize</span><span id="lstnumberx599.34" class="ltx_text ltx_font_typewriter">(0.5</span><span id="lstnumberx599.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cm</span><span id="lstnumberx599.37" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.39" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.40" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nreal</span><span id="lstnumberx599.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">func</span><span id="lstnumberx599.43" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx599.45" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.47" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx599.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.49" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx599.50" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.51" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx599.53" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.54" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx599.55" class="ltx_text ltx_font_typewriter">;\</span><span id="lstnumberx599.56" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.57" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.58" class="ltx_text ltx_lst_identifier ltx_font_typewriter">if</span><span id="lstnumberx599.59" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.60" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.63" class="ltx_text ltx_font_typewriter">&gt;=</span><span id="lstnumberx599.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.65" class="ltx_text ltx_font_typewriter">-3</span><span id="lstnumberx599.66" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.67" class="ltx_text ltx_font_typewriter">&amp;&amp;</span><span id="lstnumberx599.68" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.69" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.70" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.71" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx599.72" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.73" class="ltx_text ltx_font_typewriter">0)</span><span id="lstnumberx599.74" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.75" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx599.76" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx599.77" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.78" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx599.79" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.80" class="ltx_text ltx_font_typewriter">-2</span><span id="lstnumberx599.81" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.82" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx599.83" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.84" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.85" class="ltx_text ltx_font_typewriter">;}\</span><span id="lstnumberx599.86" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.87" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.88" class="ltx_text ltx_lst_identifier ltx_font_typewriter">if</span><span id="lstnumberx599.89" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.90" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.91" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.92" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.93" class="ltx_text ltx_font_typewriter">&gt;=</span><span id="lstnumberx599.94" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.95" class="ltx_text ltx_font_typewriter">0</span><span id="lstnumberx599.96" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.97" class="ltx_text ltx_font_typewriter">&amp;&amp;</span><span id="lstnumberx599.98" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.99" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.100" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.101" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx599.102" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.103" class="ltx_text ltx_font_typewriter">2)</span><span id="lstnumberx599.104" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.105" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx599.106" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx599.107" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.108" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx599.109" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.110" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx599.111" class="ltx_text ltx_font_typewriter">(4</span><span id="lstnumberx599.112" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.113" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx599.114" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.115" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.116" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.117" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.118" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx599.119" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.120" class="ltx_text ltx_font_typewriter">2)^2)</span><span id="lstnumberx599.121" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.122" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx599.123" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.124" class="ltx_text ltx_font_typewriter">2;}\</span><span id="lstnumberx599.125" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.126" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.127" class="ltx_text ltx_lst_identifier ltx_font_typewriter">if</span><span id="lstnumberx599.128" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.129" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.130" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.131" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.132" class="ltx_text ltx_font_typewriter">&gt;=</span><span id="lstnumberx599.133" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.134" class="ltx_text ltx_font_typewriter">2</span><span id="lstnumberx599.135" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.136" class="ltx_text ltx_font_typewriter">&amp;&amp;</span><span id="lstnumberx599.137" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.138" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.139" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.140" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx599.141" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.142" class="ltx_text ltx_font_typewriter">3)</span><span id="lstnumberx599.143" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.144" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx599.145" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx599.146" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.147" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx599.148" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.149" class="ltx_text ltx_font_typewriter">2*(</span><span id="lstnumberx599.150" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.151" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.152" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx599.153" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.154" class="ltx_text ltx_font_typewriter">2);}\</span><span id="lstnumberx599.155" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.156" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.157" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx599.158" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.159" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx599.160" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.161" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.162" class="ltx_text ltx_font_typewriter">}\</span><span id="lstnumberx599.163" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.164" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.165" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nint</span><span id="lstnumberx599.166" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.167" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx599.168" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx599.169" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.170" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.171" class="ltx_text ltx_font_typewriter">;\</span><span id="lstnumberx599.172" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.173" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.174" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nfor</span><span id="lstnumberx599.175" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.176" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.177" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx599.178" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.179" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx599.180" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.181" class="ltx_text ltx_font_typewriter">-5;</span><span id="lstnumberx599.182" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.183" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx599.184" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.185" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx599.186" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.187" class="ltx_text ltx_font_typewriter">5;</span><span id="lstnumberx599.188" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.189" class="ltx_text ltx_font_typewriter">++</span><span id="lstnumberx599.190" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx599.191" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx599.192" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.193" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx599.194" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.195" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.196" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx599.197" class="ltx_text ltx_font_typewriter">((</span><span id="lstnumberx599.198" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx599.199" class="ltx_text ltx_font_typewriter">,-5)--(</span><span id="lstnumberx599.200" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx599.201" class="ltx_text ltx_font_typewriter">,5),</span><span id="lstnumberx599.202" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gray</span><span id="lstnumberx599.203" class="ltx_text ltx_font_typewriter">(0.7));\</span><span id="lstnumberx599.204" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.205" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.206" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx599.207" class="ltx_text ltx_font_typewriter">((-5,</span><span id="lstnumberx599.208" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx599.209" class="ltx_text ltx_font_typewriter">)--(5,</span><span id="lstnumberx599.210" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx599.211" class="ltx_text ltx_font_typewriter">),</span><span id="lstnumberx599.212" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gray</span><span id="lstnumberx599.213" class="ltx_text ltx_font_typewriter">(0.7));\</span><span id="lstnumberx599.214" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.215" class="ltx_text ltx_font_typewriter">}\</span><span id="lstnumberx599.216" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.217" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.218" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx599.219" class="ltx_text ltx_font_typewriter">((-5,0)--(5,0),</span><span id="lstnumberx599.220" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Arrows</span><span id="lstnumberx599.221" class="ltx_text ltx_font_typewriter">(6));\</span><span id="lstnumberx599.222" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx599.223" class="ltx_text ltx_font_typewriter">((0,-5)--(0,5),</span><span id="lstnumberx599.224" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Arrows</span><span id="lstnumberx599.225" class="ltx_text ltx_font_typewriter">(6));\</span><span id="lstnumberx599.226" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.227" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.228" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nlabel</span><span id="lstnumberx599.229" class="ltx_text ltx_font_typewriter">("</span><span id="lstnumberx599.230" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x$</span><span id="lstnumberx599.231" class="ltx_text ltx_font_typewriter">",</span><span id="lstnumberx599.232" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.233" class="ltx_text ltx_font_typewriter">(5,0),</span><span id="lstnumberx599.234" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.235" class="ltx_text ltx_lst_identifier ltx_font_typewriter">E</span><span id="lstnumberx599.236" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.237" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nlabel</span><span id="lstnumberx599.238" class="ltx_text ltx_font_typewriter">("</span><span id="lstnumberx599.239" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y$</span><span id="lstnumberx599.240" class="ltx_text ltx_font_typewriter">",</span><span id="lstnumberx599.241" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.242" class="ltx_text ltx_font_typewriter">(0,5),</span><span id="lstnumberx599.243" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.244" class="ltx_text ltx_lst_identifier ltx_font_typewriter">N</span><span id="lstnumberx599.245" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.246" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.247" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.248" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx599.249" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.250" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx599.251" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.252" class="ltx_text ltx_lst_identifier ltx_font_typewriter">func</span><span id="lstnumberx599.253" class="ltx_text ltx_font_typewriter">,-3,3),</span><span id="lstnumberx599.254" class="ltx_text ltx_lst_identifier ltx_font_typewriter">red</span><span id="lstnumberx599.255" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.256" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.257" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.258" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nlabel</span><span id="lstnumberx599.259" class="ltx_text ltx_font_typewriter">("</span><span id="lstnumberx599.260" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx599.261" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.262" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx599.263" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.264" class="ltx_text ltx_lst_identifier ltx_font_typewriter">f</span><span id="lstnumberx599.265" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.266" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.267" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx599.268" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx599.269" class="ltx_text ltx_font_typewriter">",</span><span id="lstnumberx599.270" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.271" class="ltx_text ltx_font_typewriter">(3,-2),</span><span id="lstnumberx599.272" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.273" class="ltx_text ltx_lst_identifier ltx_font_typewriter">UnFill</span><span id="lstnumberx599.274" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.275" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.276" class="ltx_text ltx_font_typewriter">[/</span><span id="lstnumberx599.277" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asy</span><span id="lstnumberx599.278" class="ltx_text ltx_font_typewriter">]\</span><span id="lstnumberx599.279" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.280" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.281" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nWhich</span><span id="lstnumberx599.282" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.283" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx599.284" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.285" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx599.286" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.287" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx599.288" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.289" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx599.290" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.291" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx599.292" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.293" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx599.294" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.295" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx599.296" class="ltx_text ltx_lst_identifier ltx_font_typewriter">f</span><span id="lstnumberx599.297" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.298" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.299" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx599.300" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx599.301" class="ltx_text ltx_font_typewriter">?\</span><span id="lstnumberx599.302" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.303" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.304" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.305" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx599.306" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asy</span><span id="lstnumberx599.307" class="ltx_text ltx_font_typewriter">]\</span><span id="lstnumberx599.308" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nunitsize</span><span id="lstnumberx599.309" class="ltx_text ltx_font_typewriter">(0.5</span><span id="lstnumberx599.310" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.311" class="ltx_text ltx_lst_identifier ltx_font_typewriter">cm</span><span id="lstnumberx599.312" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.313" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.314" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.315" class="ltx_text ltx_lst_identifier ltx_font_typewriter">npicture</span><span id="lstnumberx599.316" class="ltx_text ltx_font_typewriter">[]</span><span id="lstnumberx599.317" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.318" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.319" class="ltx_text ltx_font_typewriter">;\</span><span id="lstnumberx599.320" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nint</span><span id="lstnumberx599.321" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.322" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx599.323" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx599.324" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.325" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.326" class="ltx_text ltx_font_typewriter">;\</span><span id="lstnumberx599.327" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.328" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.329" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nreal</span><span id="lstnumberx599.330" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.331" class="ltx_text ltx_lst_identifier ltx_font_typewriter">func</span><span id="lstnumberx599.332" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.333" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx599.334" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.335" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.336" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx599.337" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.338" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx599.339" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.340" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.341" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx599.342" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.343" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx599.344" class="ltx_text ltx_font_typewriter">;\</span><span id="lstnumberx599.345" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.346" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.347" class="ltx_text ltx_lst_identifier ltx_font_typewriter">if</span><span id="lstnumberx599.348" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.349" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.350" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.351" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.352" class="ltx_text ltx_font_typewriter">&gt;=</span><span id="lstnumberx599.353" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.354" class="ltx_text ltx_font_typewriter">-3</span><span id="lstnumberx599.355" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.356" class="ltx_text ltx_font_typewriter">&amp;&amp;</span><span id="lstnumberx599.357" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.358" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.359" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.360" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx599.361" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.362" class="ltx_text ltx_font_typewriter">0)</span><span id="lstnumberx599.363" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.364" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx599.365" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx599.366" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.367" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx599.368" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.369" class="ltx_text ltx_font_typewriter">-2</span><span id="lstnumberx599.370" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.371" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx599.372" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.373" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.374" class="ltx_text ltx_font_typewriter">;}\</span><span id="lstnumberx599.375" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.376" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.377" class="ltx_text ltx_lst_identifier ltx_font_typewriter">if</span><span id="lstnumberx599.378" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.379" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.380" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.381" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.382" class="ltx_text ltx_font_typewriter">&gt;=</span><span id="lstnumberx599.383" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.384" class="ltx_text ltx_font_typewriter">0</span><span id="lstnumberx599.385" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.386" class="ltx_text ltx_font_typewriter">&amp;&amp;</span><span id="lstnumberx599.387" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.388" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.389" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.390" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx599.391" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.392" class="ltx_text ltx_font_typewriter">2)</span><span id="lstnumberx599.393" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.394" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx599.395" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx599.396" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.397" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx599.398" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.399" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sqrt</span><span id="lstnumberx599.400" class="ltx_text ltx_font_typewriter">(4</span><span id="lstnumberx599.401" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.402" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx599.403" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.404" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.405" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.406" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.407" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx599.408" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.409" class="ltx_text ltx_font_typewriter">2)^2)</span><span id="lstnumberx599.410" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.411" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx599.412" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.413" class="ltx_text ltx_font_typewriter">2;}\</span><span id="lstnumberx599.414" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.415" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.416" class="ltx_text ltx_lst_identifier ltx_font_typewriter">if</span><span id="lstnumberx599.417" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.418" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.419" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.420" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.421" class="ltx_text ltx_font_typewriter">&gt;=</span><span id="lstnumberx599.422" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.423" class="ltx_text ltx_font_typewriter">2</span><span id="lstnumberx599.424" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.425" class="ltx_text ltx_font_typewriter">&amp;&amp;</span><span id="lstnumberx599.426" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.427" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.428" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.429" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx599.430" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.431" class="ltx_text ltx_font_typewriter">3)</span><span id="lstnumberx599.432" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.433" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx599.434" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx599.435" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.436" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx599.437" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.438" class="ltx_text ltx_font_typewriter">2*(</span><span id="lstnumberx599.439" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.440" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.441" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx599.442" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.443" class="ltx_text ltx_font_typewriter">2);}\</span><span id="lstnumberx599.444" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.445" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.446" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx599.447" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.448" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx599.449" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.450" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.451" class="ltx_text ltx_font_typewriter">}\</span><span id="lstnumberx599.452" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.453" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.454" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nreal</span><span id="lstnumberx599.455" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.456" class="ltx_text ltx_lst_identifier ltx_font_typewriter">funcb</span><span id="lstnumberx599.457" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.458" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx599.459" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.460" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.461" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx599.462" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.463" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx599.464" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.465" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.466" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx599.467" class="ltx_text ltx_font_typewriter">(-</span><span id="lstnumberx599.468" class="ltx_text ltx_lst_identifier ltx_font_typewriter">func</span><span id="lstnumberx599.469" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.470" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.471" class="ltx_text ltx_font_typewriter">));\</span><span id="lstnumberx599.472" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.473" class="ltx_text ltx_font_typewriter">}\</span><span id="lstnumberx599.474" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.475" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.476" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nreal</span><span id="lstnumberx599.477" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.478" class="ltx_text ltx_lst_identifier ltx_font_typewriter">funcd</span><span id="lstnumberx599.479" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.480" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx599.481" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.482" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.483" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx599.484" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.485" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx599.486" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.487" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.488" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx599.489" class="ltx_text ltx_font_typewriter">(-</span><span id="lstnumberx599.490" class="ltx_text ltx_lst_identifier ltx_font_typewriter">func</span><span id="lstnumberx599.491" class="ltx_text ltx_font_typewriter">(-</span><span id="lstnumberx599.492" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.493" class="ltx_text ltx_font_typewriter">));\</span><span id="lstnumberx599.494" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.495" class="ltx_text ltx_font_typewriter">}\</span><span id="lstnumberx599.496" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.497" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.498" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nreal</span><span id="lstnumberx599.499" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.500" class="ltx_text ltx_lst_identifier ltx_font_typewriter">funce</span><span id="lstnumberx599.501" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.502" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx599.503" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.504" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.505" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx599.506" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.507" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx599.508" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.509" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.510" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx599.511" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.512" class="ltx_text ltx_lst_identifier ltx_font_typewriter">func</span><span id="lstnumberx599.513" class="ltx_text ltx_font_typewriter">(-</span><span id="lstnumberx599.514" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.515" class="ltx_text ltx_font_typewriter">));\</span><span id="lstnumberx599.516" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.517" class="ltx_text ltx_font_typewriter">}\</span><span id="lstnumberx599.518" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.519" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.520" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nfor</span><span id="lstnumberx599.521" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.522" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.523" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.524" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.525" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx599.526" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.527" class="ltx_text ltx_font_typewriter">1;</span><span id="lstnumberx599.528" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.529" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.530" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.531" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx599.532" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.533" class="ltx_text ltx_font_typewriter">5;</span><span id="lstnumberx599.534" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.535" class="ltx_text ltx_font_typewriter">++</span><span id="lstnumberx599.536" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.537" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx599.538" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.539" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx599.540" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.541" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.542" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.543" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx599.544" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.545" class="ltx_text ltx_font_typewriter">]</span><span id="lstnumberx599.546" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.547" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx599.548" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.549" class="ltx_text ltx_lst_identifier ltx_font_typewriter">new</span><span id="lstnumberx599.550" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.551" class="ltx_text ltx_lst_identifier ltx_font_typewriter">picture</span><span id="lstnumberx599.552" class="ltx_text ltx_font_typewriter">;\</span><span id="lstnumberx599.553" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.554" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.555" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx599.556" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.557" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.558" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx599.559" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.560" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx599.561" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.562" class="ltx_text ltx_font_typewriter">-5;</span><span id="lstnumberx599.563" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.564" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx599.565" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.566" class="ltx_text ltx_font_typewriter">&lt;=</span><span id="lstnumberx599.567" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.568" class="ltx_text ltx_font_typewriter">5;</span><span id="lstnumberx599.569" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.570" class="ltx_text ltx_font_typewriter">++</span><span id="lstnumberx599.571" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx599.572" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx599.573" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.574" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx599.575" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.576" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx599.577" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx599.578" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.579" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.580" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx599.581" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.582" class="ltx_text ltx_font_typewriter">],(</span><span id="lstnumberx599.583" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx599.584" class="ltx_text ltx_font_typewriter">,-5)--(</span><span id="lstnumberx599.585" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx599.586" class="ltx_text ltx_font_typewriter">,5),</span><span id="lstnumberx599.587" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gray</span><span id="lstnumberx599.588" class="ltx_text ltx_font_typewriter">(0.7));\</span><span id="lstnumberx599.589" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.590" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx599.591" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx599.592" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.593" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.594" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx599.595" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.596" class="ltx_text ltx_font_typewriter">],(-5,</span><span id="lstnumberx599.597" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx599.598" class="ltx_text ltx_font_typewriter">)--(5,</span><span id="lstnumberx599.599" class="ltx_text ltx_lst_identifier ltx_font_typewriter">i</span><span id="lstnumberx599.600" class="ltx_text ltx_font_typewriter">),</span><span id="lstnumberx599.601" class="ltx_text ltx_lst_identifier ltx_font_typewriter">gray</span><span id="lstnumberx599.602" class="ltx_text ltx_font_typewriter">(0.7));\</span><span id="lstnumberx599.603" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.604" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.605" class="ltx_text ltx_font_typewriter">}\</span><span id="lstnumberx599.606" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.607" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.608" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx599.609" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.610" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.611" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx599.612" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.613" class="ltx_text ltx_font_typewriter">],(-5,0)--(5,0),</span><span id="lstnumberx599.614" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Arrows</span><span id="lstnumberx599.615" class="ltx_text ltx_font_typewriter">(6));\</span><span id="lstnumberx599.616" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.617" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.618" class="ltx_text ltx_lst_identifier ltx_font_typewriter">draw</span><span id="lstnumberx599.619" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.620" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.621" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx599.622" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.623" class="ltx_text ltx_font_typewriter">],(0,-5)--(0,5),</span><span id="lstnumberx599.624" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Arrows</span><span id="lstnumberx599.625" class="ltx_text ltx_font_typewriter">(6));\</span><span id="lstnumberx599.626" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.627" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.628" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.629" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.630" class="ltx_text ltx_lst_identifier ltx_font_typewriter">label</span><span id="lstnumberx599.631" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.632" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.633" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx599.634" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.635" class="ltx_text ltx_font_typewriter">],"</span><span id="lstnumberx599.636" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x$</span><span id="lstnumberx599.637" class="ltx_text ltx_font_typewriter">",</span><span id="lstnumberx599.638" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.639" class="ltx_text ltx_font_typewriter">(5,0),</span><span id="lstnumberx599.640" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.641" class="ltx_text ltx_lst_identifier ltx_font_typewriter">E</span><span id="lstnumberx599.642" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.643" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.644" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx599.645" class="ltx_text ltx_lst_identifier ltx_font_typewriter">label</span><span id="lstnumberx599.646" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.647" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.648" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx599.649" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.650" class="ltx_text ltx_font_typewriter">],"</span><span id="lstnumberx599.651" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y$</span><span id="lstnumberx599.652" class="ltx_text ltx_font_typewriter">",</span><span id="lstnumberx599.653" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.654" class="ltx_text ltx_font_typewriter">(0,5),</span><span id="lstnumberx599.655" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.656" class="ltx_text ltx_lst_identifier ltx_font_typewriter">N</span><span id="lstnumberx599.657" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.658" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.659" class="ltx_text ltx_font_typewriter">}\</span><span id="lstnumberx599.660" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.661" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.662" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx599.663" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.664" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.665" class="ltx_text ltx_font_typewriter">[1],(-5,3)--(-2,0),</span><span id="lstnumberx599.666" class="ltx_text ltx_lst_identifier ltx_font_typewriter">red</span><span id="lstnumberx599.667" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.668" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx599.669" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.670" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.671" class="ltx_text ltx_font_typewriter">[1],</span><span id="lstnumberx599.672" class="ltx_text ltx_lst_identifier ltx_font_typewriter">arc</span><span id="lstnumberx599.673" class="ltx_text ltx_font_typewriter">((-2,2),2,270,360),</span><span id="lstnumberx599.674" class="ltx_text ltx_lst_identifier ltx_font_typewriter">red</span><span id="lstnumberx599.675" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.676" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx599.677" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.678" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.679" class="ltx_text ltx_font_typewriter">[1],(0,2)--(2,4),</span><span id="lstnumberx599.680" class="ltx_text ltx_lst_identifier ltx_font_typewriter">red</span><span id="lstnumberx599.681" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.682" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx599.683" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.684" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.685" class="ltx_text ltx_font_typewriter">[2],</span><span id="lstnumberx599.686" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx599.687" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.688" class="ltx_text ltx_lst_identifier ltx_font_typewriter">funcb</span><span id="lstnumberx599.689" class="ltx_text ltx_font_typewriter">,-3,3),</span><span id="lstnumberx599.690" class="ltx_text ltx_lst_identifier ltx_font_typewriter">red</span><span id="lstnumberx599.691" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.692" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx599.693" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.694" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.695" class="ltx_text ltx_font_typewriter">[3],(-3,2)--(-2,0),</span><span id="lstnumberx599.696" class="ltx_text ltx_lst_identifier ltx_font_typewriter">red</span><span id="lstnumberx599.697" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.698" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx599.699" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.700" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.701" class="ltx_text ltx_font_typewriter">[3],</span><span id="lstnumberx599.702" class="ltx_text ltx_lst_identifier ltx_font_typewriter">arc</span><span id="lstnumberx599.703" class="ltx_text ltx_font_typewriter">((-2,-2),2,0,90),</span><span id="lstnumberx599.704" class="ltx_text ltx_lst_identifier ltx_font_typewriter">red</span><span id="lstnumberx599.705" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.706" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx599.707" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.708" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.709" class="ltx_text ltx_font_typewriter">[3],(0,-2)--(3,-5),</span><span id="lstnumberx599.710" class="ltx_text ltx_lst_identifier ltx_font_typewriter">red</span><span id="lstnumberx599.711" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.712" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx599.713" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.714" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.715" class="ltx_text ltx_font_typewriter">[4],</span><span id="lstnumberx599.716" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx599.717" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.718" class="ltx_text ltx_lst_identifier ltx_font_typewriter">funcd</span><span id="lstnumberx599.719" class="ltx_text ltx_font_typewriter">,-3,3),</span><span id="lstnumberx599.720" class="ltx_text ltx_lst_identifier ltx_font_typewriter">red</span><span id="lstnumberx599.721" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.722" class="ltx_text ltx_lst_identifier ltx_font_typewriter">ndraw</span><span id="lstnumberx599.723" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.724" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.725" class="ltx_text ltx_font_typewriter">[5],</span><span id="lstnumberx599.726" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx599.727" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.728" class="ltx_text ltx_lst_identifier ltx_font_typewriter">funce</span><span id="lstnumberx599.729" class="ltx_text ltx_font_typewriter">,-3,3),</span><span id="lstnumberx599.730" class="ltx_text ltx_lst_identifier ltx_font_typewriter">red</span><span id="lstnumberx599.731" class="ltx_text ltx_font_typewriter">);\</span><span id="lstnumberx599.732" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.733" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.734" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nlabel</span><span id="lstnumberx599.735" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.736" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.737" class="ltx_text ltx_font_typewriter">[1],</span><span id="lstnumberx599.738" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.739" class="ltx_text ltx_font_typewriter">"</span><span id="lstnumberx599.740" class="ltx_text ltx_lst_identifier ltx_font_typewriter">A</span><span id="lstnumberx599.741" class="ltx_text ltx_font_typewriter">",</span><span id="lstnumberx599.742" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.743" class="ltx_text ltx_font_typewriter">(0,-6));\</span><span id="lstnumberx599.744" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nlabel</span><span id="lstnumberx599.745" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.746" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.747" class="ltx_text ltx_font_typewriter">[2],</span><span id="lstnumberx599.748" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.749" class="ltx_text ltx_font_typewriter">"</span><span id="lstnumberx599.750" class="ltx_text ltx_lst_identifier ltx_font_typewriter">B</span><span id="lstnumberx599.751" class="ltx_text ltx_font_typewriter">",</span><span id="lstnumberx599.752" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.753" class="ltx_text ltx_font_typewriter">(0,-6));\</span><span id="lstnumberx599.754" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nlabel</span><span id="lstnumberx599.755" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.756" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.757" class="ltx_text ltx_font_typewriter">[3],</span><span id="lstnumberx599.758" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.759" class="ltx_text ltx_font_typewriter">"</span><span id="lstnumberx599.760" class="ltx_text ltx_lst_identifier ltx_font_typewriter">C</span><span id="lstnumberx599.761" class="ltx_text ltx_font_typewriter">",</span><span id="lstnumberx599.762" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.763" class="ltx_text ltx_font_typewriter">(0,-6));\</span><span id="lstnumberx599.764" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nlabel</span><span id="lstnumberx599.765" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.766" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.767" class="ltx_text ltx_font_typewriter">[4],</span><span id="lstnumberx599.768" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.769" class="ltx_text ltx_font_typewriter">"</span><span id="lstnumberx599.770" class="ltx_text ltx_lst_identifier ltx_font_typewriter">D</span><span id="lstnumberx599.771" class="ltx_text ltx_font_typewriter">",</span><span id="lstnumberx599.772" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.773" class="ltx_text ltx_font_typewriter">(0,-6));\</span><span id="lstnumberx599.774" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nlabel</span><span id="lstnumberx599.775" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.776" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.777" class="ltx_text ltx_font_typewriter">[5],</span><span id="lstnumberx599.778" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.779" class="ltx_text ltx_font_typewriter">"</span><span id="lstnumberx599.780" class="ltx_text ltx_lst_identifier ltx_font_typewriter">E</span><span id="lstnumberx599.781" class="ltx_text ltx_font_typewriter">",</span><span id="lstnumberx599.782" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.783" class="ltx_text ltx_font_typewriter">(0,-6));\</span><span id="lstnumberx599.784" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.785" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.786" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nadd</span><span id="lstnumberx599.787" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.788" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.789" class="ltx_text ltx_font_typewriter">[1]);\</span><span id="lstnumberx599.790" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nadd</span><span id="lstnumberx599.791" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.792" class="ltx_text ltx_lst_identifier ltx_font_typewriter">shift</span><span id="lstnumberx599.793" class="ltx_text ltx_font_typewriter">((12,0))*(</span><span id="lstnumberx599.794" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.795" class="ltx_text ltx_font_typewriter">[2]));\</span><span id="lstnumberx599.796" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nadd</span><span id="lstnumberx599.797" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.798" class="ltx_text ltx_lst_identifier ltx_font_typewriter">shift</span><span id="lstnumberx599.799" class="ltx_text ltx_font_typewriter">((24,0))*(</span><span id="lstnumberx599.800" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.801" class="ltx_text ltx_font_typewriter">[3]));\</span><span id="lstnumberx599.802" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nadd</span><span id="lstnumberx599.803" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.804" class="ltx_text ltx_lst_identifier ltx_font_typewriter">shift</span><span id="lstnumberx599.805" class="ltx_text ltx_font_typewriter">((6,-12))*(</span><span id="lstnumberx599.806" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.807" class="ltx_text ltx_font_typewriter">[4]));\</span><span id="lstnumberx599.808" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nadd</span><span id="lstnumberx599.809" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.810" class="ltx_text ltx_lst_identifier ltx_font_typewriter">shift</span><span id="lstnumberx599.811" class="ltx_text ltx_font_typewriter">((18,-12))*(</span><span id="lstnumberx599.812" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graf</span><span id="lstnumberx599.813" class="ltx_text ltx_font_typewriter">[5]));\</span><span id="lstnumberx599.814" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.815" class="ltx_text ltx_font_typewriter">[/</span><span id="lstnumberx599.816" class="ltx_text ltx_lst_identifier ltx_font_typewriter">asy</span><span id="lstnumberx599.817" class="ltx_text ltx_font_typewriter">]\</span><span id="lstnumberx599.818" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx599.819" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx599.820" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nEnter</span><span id="lstnumberx599.821" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.822" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx599.823" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.824" class="ltx_text ltx_lst_identifier ltx_font_typewriter">letter</span><span id="lstnumberx599.825" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.826" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx599.827" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.828" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx599.829" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.830" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx599.831" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.832" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx599.833" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.834" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx599.835" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.836" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx599.837" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx599.838" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx599.839" class="ltx_text ltx_lst_identifier ltx_font_typewriter">f</span><span id="lstnumberx599.840" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx599.841" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx599.842" class="ltx_text ltx_font_typewriter">).</span><span id="lstnumberx599.843" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx599.844" class="ltx_text ltx_font_typewriter">’</span>
</div>
<div id="lstnumberx600" class="ltx_listingline">
<span id="lstnumberx600.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solution</span><span id="lstnumberx600.2" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx601" class="ltx_listingline">
<span id="lstnumberx601.2" class="ltx_text ltx_font_typewriter" style="background-color:#F47874;">The correct graph is the one that reflects the original graph <math id="lstnumberx601.1.1.m1.1" class="ltx_Math" alttext="y=f(x)" display="inline"><semantics id="lstnumberx601.1.1.m1.1a"><mrow id="lstnumberx601.1.1.m1.1.2" xref="lstnumberx601.1.1.m1.1.2.cmml"><mi mathbackground="#F47874" id="lstnumberx601.1.1.m1.1.2.2" xref="lstnumberx601.1.1.m1.1.2.2.cmml">y</mi><mo mathbackground="#F47874" id="lstnumberx601.1.1.m1.1.2.1" xref="lstnumberx601.1.1.m1.1.2.1.cmml">=</mo><mrow id="lstnumberx601.1.1.m1.1.2.3" xref="lstnumberx601.1.1.m1.1.2.3.cmml"><mi mathbackground="#F47874" id="lstnumberx601.1.1.m1.1.2.3.2" xref="lstnumberx601.1.1.m1.1.2.3.2.cmml">f</mi><mo lspace="0em" rspace="0em" id="lstnumberx601.1.1.m1.1.2.3.1" xref="lstnumberx601.1.1.m1.1.2.3.1.cmml">​</mo><mrow id="lstnumberx601.1.1.m1.1.2.3.3.2" xref="lstnumberx601.1.1.m1.1.2.3.cmml"><mo mathbackground="#F47874" stretchy="false" id="lstnumberx601.1.1.m1.1.2.3.3.2.1" xref="lstnumberx601.1.1.m1.1.2.3.cmml">(</mo><mi mathbackground="#F47874" id="lstnumberx601.1.1.m1.1.1" xref="lstnumberx601.1.1.m1.1.1.cmml">x</mi><mo mathbackground="#F47874" stretchy="false" id="lstnumberx601.1.1.m1.1.2.3.3.2.2" xref="lstnumberx601.1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="lstnumberx601.1.1.m1.1b"><apply id="lstnumberx601.1.1.m1.1.2.cmml" xref="lstnumberx601.1.1.m1.1.2"><eq id="lstnumberx601.1.1.m1.1.2.1.cmml" xref="lstnumberx601.1.1.m1.1.2.1"></eq><ci id="lstnumberx601.1.1.m1.1.2.2.cmml" xref="lstnumberx601.1.1.m1.1.2.2">𝑦</ci><apply id="lstnumberx601.1.1.m1.1.2.3.cmml" xref="lstnumberx601.1.1.m1.1.2.3"><times id="lstnumberx601.1.1.m1.1.2.3.1.cmml" xref="lstnumberx601.1.1.m1.1.2.3.1"></times><ci id="lstnumberx601.1.1.m1.1.2.3.2.cmml" xref="lstnumberx601.1.1.m1.1.2.3.2">𝑓</ci><ci id="lstnumberx601.1.1.m1.1.1.cmml" xref="lstnumberx601.1.1.m1.1.1">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="lstnumberx601.1.1.m1.1c">y=f(x)</annotation></semantics></math> about the <math id="lstnumberx601.2.2.m2.1" class="ltx_Math" alttext="y" display="inline"><semantics id="lstnumberx601.2.2.m2.1a"><mi mathbackground="#F47874" id="lstnumberx601.2.2.m2.1.1" xref="lstnumberx601.2.2.m2.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="lstnumberx601.2.2.m2.1b"><ci id="lstnumberx601.2.2.m2.1.1.cmml" xref="lstnumberx601.2.2.m2.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="lstnumberx601.2.2.m2.1c">y</annotation></semantics></math>-axis. This is graph D.</span>
</div>
<div id="lstnumberx602" class="ltx_listingline">
</div>
<div id="lstnumberx603" class="ltx_listingline">
<span id="lstnumberx603.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">So</span><span id="lstnumberx603.2" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx603.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx603.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx603.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx603.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">answer</span><span id="lstnumberx603.7" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx603.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx603.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx603.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx603.11" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx603.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx603.13" class="ltx_text ltx_font_typewriter">{\</span><span id="lstnumberx603.14" class="ltx_text ltx_lst_identifier ltx_font_typewriter">text</span><span id="lstnumberx603.15" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx603.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">D</span><span id="lstnumberx603.17" class="ltx_text ltx_font_typewriter">}}</span><span id="lstnumberx603.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx603.19" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx604" class="ltx_listingline">
</div>
<div id="lstnumberx605" class="ltx_listingline">
<span id="lstnumberx605.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Execution</span><span id="lstnumberx605.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx605.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx605.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">None</span>
</div>
<div id="lstnumberx606" class="ltx_listingline">
<span id="lstnumberx606.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Prediction</span><span id="lstnumberx606.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx606.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx606.4" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx606.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">D</span><span id="lstnumberx606.6" class="ltx_text ltx_font_typewriter">’</span>
</div>
<div id="lstnumberx607" class="ltx_listingline">
<span id="lstnumberx607.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Ground</span><span id="lstnumberx607.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx607.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Truth</span><span id="lstnumberx607.4" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx607.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx607.6" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx607.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">B</span><span id="lstnumberx607.8" class="ltx_text ltx_font_typewriter">’</span>
</div>
<div id="lstnumberx608" class="ltx_listingline">
<span id="lstnumberx608.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Score</span><span id="lstnumberx608.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx608.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx608.4" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx608.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">False</span><span id="lstnumberx608.6" class="ltx_text ltx_font_typewriter">]</span>
</div>
<div id="lstnumberx609" class="ltx_listingline">
<span id="lstnumberx609.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Reference</span><span id="lstnumberx609.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">CoT</span><span id="lstnumberx609.4" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx609.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.6" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx609.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx609.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx609.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx609.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx609.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.15" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx609.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.17" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx609.18" class="ltx_text ltx_lst_identifier ltx_font_typewriter">f</span><span id="lstnumberx609.19" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx609.20" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx609.21" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx609.22" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx609.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx609.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx609.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">reflection</span><span id="lstnumberx609.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx609.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx609.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx609.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.36" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx609.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.38" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx609.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.40" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx609.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.42" class="ltx_text ltx_lst_identifier ltx_font_typewriter">f</span><span id="lstnumberx609.43" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx609.44" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx609.45" class="ltx_text ltx_font_typewriter">)</span><span id="lstnumberx609.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx609.47" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.48" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx609.49" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.50" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx609.51" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x$</span><span id="lstnumberx609.53" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx609.54" class="ltx_text ltx_lst_identifier ltx_font_typewriter">axis</span><span id="lstnumberx609.55" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx609.56" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx609.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx609.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.59" class="ltx_text ltx_lst_identifier ltx_font_typewriter">correct</span><span id="lstnumberx609.60" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.61" class="ltx_text ltx_lst_identifier ltx_font_typewriter">graph</span><span id="lstnumberx609.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.63" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is</span><span id="lstnumberx609.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx609.65" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx609.66" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx609.67" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx609.68" class="ltx_text ltx_font_typewriter">{\\</span><span id="lstnumberx609.69" class="ltx_text ltx_lst_identifier ltx_font_typewriter">text</span><span id="lstnumberx609.70" class="ltx_text ltx_font_typewriter">{</span><span id="lstnumberx609.71" class="ltx_text ltx_lst_identifier ltx_font_typewriter">B</span><span id="lstnumberx609.72" class="ltx_text ltx_font_typewriter">}}.</span><span id="lstnumberx609.73" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx609.74" class="ltx_text ltx_font_typewriter">’</span>
</div>
</div>
</figure>
<figure id="LST12" class="ltx_float ltx_lstlisting">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float">Listing 12: </span>Failure case for <span id="LST12.3.1" class="ltx_text ltx_font_smallcaps">ToRA</span>: <span id="LST12.4.2" class="ltx_text ltx_font_bold">False Negative</span>.</figcaption>
<div id="LST12.5" class="ltx_listing ltx_lstlisting ltx_framed ltx_framed_topbottom ltx_listing">
<div class="ltx_listing_data"><a href="data:text/plain;base64,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" download="section/cases/failure/8_false_negative.md">⬇</a></div>
<div id="lstnumberx610" class="ltx_listingline">
<span id="lstnumberx610.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Idx</span><span id="lstnumberx610.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx610.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx610.4" class="ltx_text ltx_font_typewriter">2228</span>
</div>
<div id="lstnumberx611" class="ltx_listingline">
<span id="lstnumberx611.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Type</span><span id="lstnumberx611.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx611.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx611.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Intermediate</span><span id="lstnumberx611.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx611.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Algebra</span>
</div>
<div id="lstnumberx612" class="ltx_listingline">
<span id="lstnumberx612.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Level</span><span id="lstnumberx612.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx612.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx612.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Level</span><span id="lstnumberx612.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx612.6" class="ltx_text ltx_font_typewriter">3</span>
</div>
<div id="lstnumberx613" class="ltx_listingline">
<span id="lstnumberx613.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Question</span><span id="lstnumberx613.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx613.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.4" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx613.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Find</span><span id="lstnumberx613.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">all</span><span id="lstnumberx613.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx613.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx613.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots</span><span id="lstnumberx613.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx613.16" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx613.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx613.18" class="ltx_text ltx_font_typewriter">\\[(12</span><span id="lstnumberx613.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx613.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.21" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx613.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.23" class="ltx_text ltx_font_typewriter">1)(6</span><span id="lstnumberx613.24" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx613.25" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.26" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx613.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.28" class="ltx_text ltx_font_typewriter">1)(4</span><span id="lstnumberx613.29" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx613.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.31" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx613.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.33" class="ltx_text ltx_font_typewriter">1)(3</span><span id="lstnumberx613.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx613.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.36" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx613.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.38" class="ltx_text ltx_font_typewriter">1)</span><span id="lstnumberx613.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.40" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx613.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.42" class="ltx_text ltx_font_typewriter">5.\\]</span><span id="lstnumberx613.43" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Enter</span><span id="lstnumberx613.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.45" class="ltx_text ltx_lst_identifier ltx_font_typewriter">all</span><span id="lstnumberx613.46" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.47" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx613.48" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.49" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx613.50" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.51" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots</span><span id="lstnumberx613.52" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx613.53" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.54" class="ltx_text ltx_lst_identifier ltx_font_typewriter">separated</span><span id="lstnumberx613.55" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.56" class="ltx_text ltx_lst_identifier ltx_font_typewriter">by</span><span id="lstnumberx613.57" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx613.58" class="ltx_text ltx_lst_identifier ltx_font_typewriter">commas</span><span id="lstnumberx613.59" class="ltx_text ltx_font_typewriter">.’</span>
</div>
<div id="lstnumberx614" class="ltx_listingline">
<span id="lstnumberx614.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Solution</span><span id="lstnumberx614.2" class="ltx_text ltx_font_typewriter">:</span>
</div>
<div id="lstnumberx615" class="ltx_listingline">
<span id="lstnumberx615.1" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx615.2" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx615.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">python</span>
</div>
<div id="lstnumberx616" class="ltx_listingline">
<span id="lstnumberx616.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">from</span><span id="lstnumberx616.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx616.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">sympy</span><span id="lstnumberx616.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx616.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">import</span><span id="lstnumberx616.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx616.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span><span id="lstnumberx616.8" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx616.9" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx616.10" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve</span><span id="lstnumberx616.11" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx616.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx616.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">simplify</span>
</div>
<div id="lstnumberx617" class="ltx_listingline">
</div>
<div id="lstnumberx618" class="ltx_listingline">
<span id="lstnumberx618.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">def</span><span id="lstnumberx618.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx618.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">find_real_roots</span><span id="lstnumberx618.4" class="ltx_text ltx_font_typewriter">():</span>
</div>
<div id="lstnumberx619" class="ltx_listingline">
<span id="lstnumberx619.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx619.2" class="ltx_text ltx_font_typewriter">"""</span><span id="lstnumberx619.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Find</span><span id="lstnumberx619.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx619.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">all</span><span id="lstnumberx619.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx619.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx619.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx619.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx619.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx619.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots</span><span id="lstnumberx619.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx619.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span>
</div>
<div id="lstnumberx620" class="ltx_listingline">
<span id="lstnumberx620.1" class="ltx_text ltx_font_typewriter">\[(12</span><span id="lstnumberx620.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx620.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx620.4" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx620.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx620.6" class="ltx_text ltx_font_typewriter">1)(6</span><span id="lstnumberx620.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx620.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx620.9" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx620.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx620.11" class="ltx_text ltx_font_typewriter">1)(4</span><span id="lstnumberx620.12" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx620.13" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx620.14" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx620.15" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx620.16" class="ltx_text ltx_font_typewriter">1)(3</span><span id="lstnumberx620.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx620.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx620.19" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx620.20" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx620.21" class="ltx_text ltx_font_typewriter">1)</span><span id="lstnumberx620.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx620.23" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx620.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx620.25" class="ltx_text ltx_font_typewriter">5.\]</span><span id="lstnumberx620.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Enter</span><span id="lstnumberx620.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx620.28" class="ltx_text ltx_lst_identifier ltx_font_typewriter">all</span><span id="lstnumberx620.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx620.30" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx620.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx620.32" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx620.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx620.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots</span><span id="lstnumberx620.35" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx620.36" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx620.37" class="ltx_text ltx_lst_identifier ltx_font_typewriter">separated</span><span id="lstnumberx620.38" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx620.39" class="ltx_text ltx_lst_identifier ltx_font_typewriter">by</span><span id="lstnumberx620.40" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx620.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">commas</span><span id="lstnumberx620.42" class="ltx_text ltx_font_typewriter">."""</span>
</div>
<div id="lstnumberx621" class="ltx_listingline">
<span id="lstnumberx621.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx621.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx621.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx621.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx621.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx621.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">symbols</span><span id="lstnumberx621.7" class="ltx_text ltx_font_typewriter">(’</span><span id="lstnumberx621.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx621.9" class="ltx_text ltx_font_typewriter">’)</span>
</div>
<div id="lstnumberx622" class="ltx_listingline">
<span id="lstnumberx622.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx622.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equation</span><span id="lstnumberx622.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx622.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.6" class="ltx_text ltx_font_typewriter">(12*</span><span id="lstnumberx622.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx622.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.9" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx622.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.11" class="ltx_text ltx_font_typewriter">1)</span><span id="lstnumberx622.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.13" class="ltx_text ltx_font_typewriter">*</span><span id="lstnumberx622.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.15" class="ltx_text ltx_font_typewriter">(6*</span><span id="lstnumberx622.16" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx622.17" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.18" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx622.19" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.20" class="ltx_text ltx_font_typewriter">1)</span><span id="lstnumberx622.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.22" class="ltx_text ltx_font_typewriter">*</span><span id="lstnumberx622.23" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.24" class="ltx_text ltx_font_typewriter">(4*</span><span id="lstnumberx622.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx622.26" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.27" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx622.28" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.29" class="ltx_text ltx_font_typewriter">1)</span><span id="lstnumberx622.30" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.31" class="ltx_text ltx_font_typewriter">*</span><span id="lstnumberx622.32" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.33" class="ltx_text ltx_font_typewriter">(3*</span><span id="lstnumberx622.34" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx622.35" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.36" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx622.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.38" class="ltx_text ltx_font_typewriter">1)</span><span id="lstnumberx622.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.40" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx622.41" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx622.42" class="ltx_text ltx_font_typewriter">5</span>
</div>
<div id="lstnumberx623" class="ltx_listingline">
<span id="lstnumberx623.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx623.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots</span><span id="lstnumberx623.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx623.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx623.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx623.6" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solve</span><span id="lstnumberx623.7" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx623.8" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equation</span><span id="lstnumberx623.9" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx623.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx623.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx623.12" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx624" class="ltx_listingline">
<span id="lstnumberx624.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx624.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real_roots</span><span id="lstnumberx624.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx624.4" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx624.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx624.6" class="ltx_text ltx_font_typewriter">[</span><span id="lstnumberx624.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">root</span><span id="lstnumberx624.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx624.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">for</span><span id="lstnumberx624.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx624.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">root</span><span id="lstnumberx624.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx624.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">in</span><span id="lstnumberx624.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx624.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots</span><span id="lstnumberx624.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx624.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">if</span><span id="lstnumberx624.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx624.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">root</span><span id="lstnumberx624.20" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx624.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">is_real</span><span id="lstnumberx624.22" class="ltx_text ltx_font_typewriter">]</span>
</div>
<div id="lstnumberx625" class="ltx_listingline">
</div>
<div id="lstnumberx626" class="ltx_listingline">
<span id="lstnumberx626.1" class="ltx_text ltx_lst_space ltx_font_typewriter">    </span><span id="lstnumberx626.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">return</span><span id="lstnumberx626.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx626.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real_roots</span>
</div>
<div id="lstnumberx627" class="ltx_listingline">
</div>
<div id="lstnumberx628" class="ltx_listingline">
<span id="lstnumberx628.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx628.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx628.3" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx628.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx628.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">find_real_roots</span><span id="lstnumberx628.6" class="ltx_text ltx_font_typewriter">()</span>
</div>
<div id="lstnumberx629" class="ltx_listingline">
<span id="lstnumberx629.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">print</span><span id="lstnumberx629.2" class="ltx_text ltx_font_typewriter">(</span><span id="lstnumberx629.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">result</span><span id="lstnumberx629.4" class="ltx_text ltx_font_typewriter">)</span>
</div>
<div id="lstnumberx630" class="ltx_listingline">
<span id="lstnumberx630.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx631" class="ltx_listingline">
<span id="lstnumberx631.1" class="ltx_text ltx_font_typewriter">‘‘‘</span><span id="lstnumberx631.2" class="ltx_text ltx_lst_identifier ltx_font_typewriter">output</span>
</div>
<div id="lstnumberx632" class="ltx_listingline">
<span id="lstnumberx632.1" class="ltx_text ltx_font_typewriter">[-1/12,</span><span id="lstnumberx632.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx632.3" class="ltx_text ltx_font_typewriter">1/2]</span>
</div>
<div id="lstnumberx633" class="ltx_listingline">
<span id="lstnumberx633.1" class="ltx_text ltx_font_typewriter">‘‘‘</span>
</div>
<div id="lstnumberx634" class="ltx_listingline">
<span id="lstnumberx634.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">The</span><span id="lstnumberx634.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx634.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx634.4" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx634.5" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots</span><span id="lstnumberx634.6" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx634.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">of</span><span id="lstnumberx634.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx634.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx634.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx634.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">equation</span><span id="lstnumberx634.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx634.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx634.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx634.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx634.16" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx634.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx634.18" class="ltx_text ltx_font_typewriter">{-\</span><span id="lstnumberx634.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx634.20" class="ltx_text ltx_font_typewriter">{1}{12},</span><span id="lstnumberx634.21" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx634.22" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx634.23" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx634.24" class="ltx_text ltx_font_typewriter">{1}{2}}</span><span id="lstnumberx634.25" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx634.26" class="ltx_text ltx_font_typewriter">.</span>
</div>
<div id="lstnumberx635" class="ltx_listingline">
</div>
<div id="lstnumberx636" class="ltx_listingline">
<span id="lstnumberx636.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Execution</span><span id="lstnumberx636.2" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx636.3" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx636.4" class="ltx_text ltx_lst_identifier ltx_font_typewriter">None</span>
</div>
<div id="lstnumberx637" class="ltx_listingline">
<span id="lstnumberx637.2" class="ltx_text ltx_font_typewriter" style="background-color:#F47874;">Prediction: ’<math id="lstnumberx637.1.1.m1.2" class="ltx_Math" alttext="-\frac{1}{12},\frac{1}{2}" display="inline"><semantics id="lstnumberx637.1.1.m1.2a"><mrow id="lstnumberx637.1.1.m1.2.2.1" xref="lstnumberx637.1.1.m1.2.2.2.cmml"><mrow id="lstnumberx637.1.1.m1.2.2.1.1" xref="lstnumberx637.1.1.m1.2.2.1.1.cmml"><mo mathbackground="#F47874" id="lstnumberx637.1.1.m1.2.2.1.1a" xref="lstnumberx637.1.1.m1.2.2.1.1.cmml">−</mo><mfrac id="lstnumberx637.1.1.m1.2.2.1.1.2" xref="lstnumberx637.1.1.m1.2.2.1.1.2.cmml"><mn mathbackground="#F47874" id="lstnumberx637.1.1.m1.2.2.1.1.2.2" xref="lstnumberx637.1.1.m1.2.2.1.1.2.2.cmml">1</mn><mn mathbackground="#F47874" id="lstnumberx637.1.1.m1.2.2.1.1.2.3" xref="lstnumberx637.1.1.m1.2.2.1.1.2.3.cmml">12</mn></mfrac></mrow><mo mathbackground="#F47874" id="lstnumberx637.1.1.m1.2.2.1.2" xref="lstnumberx637.1.1.m1.2.2.2.cmml">,</mo><mfrac id="lstnumberx637.1.1.m1.1.1" xref="lstnumberx637.1.1.m1.1.1.cmml"><mn mathbackground="#F47874" id="lstnumberx637.1.1.m1.1.1.2" xref="lstnumberx637.1.1.m1.1.1.2.cmml">1</mn><mn mathbackground="#F47874" id="lstnumberx637.1.1.m1.1.1.3" xref="lstnumberx637.1.1.m1.1.1.3.cmml">2</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="lstnumberx637.1.1.m1.2b"><list id="lstnumberx637.1.1.m1.2.2.2.cmml" xref="lstnumberx637.1.1.m1.2.2.1"><apply id="lstnumberx637.1.1.m1.2.2.1.1.cmml" xref="lstnumberx637.1.1.m1.2.2.1.1"><minus id="lstnumberx637.1.1.m1.2.2.1.1.1.cmml" xref="lstnumberx637.1.1.m1.2.2.1.1"></minus><apply id="lstnumberx637.1.1.m1.2.2.1.1.2.cmml" xref="lstnumberx637.1.1.m1.2.2.1.1.2"><divide id="lstnumberx637.1.1.m1.2.2.1.1.2.1.cmml" xref="lstnumberx637.1.1.m1.2.2.1.1.2"></divide><cn type="integer" id="lstnumberx637.1.1.m1.2.2.1.1.2.2.cmml" xref="lstnumberx637.1.1.m1.2.2.1.1.2.2">1</cn><cn type="integer" id="lstnumberx637.1.1.m1.2.2.1.1.2.3.cmml" xref="lstnumberx637.1.1.m1.2.2.1.1.2.3">12</cn></apply></apply><apply id="lstnumberx637.1.1.m1.1.1.cmml" xref="lstnumberx637.1.1.m1.1.1"><divide id="lstnumberx637.1.1.m1.1.1.1.cmml" xref="lstnumberx637.1.1.m1.1.1"></divide><cn type="integer" id="lstnumberx637.1.1.m1.1.1.2.cmml" xref="lstnumberx637.1.1.m1.1.1.2">1</cn><cn type="integer" id="lstnumberx637.1.1.m1.1.1.3.cmml" xref="lstnumberx637.1.1.m1.1.1.3">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="lstnumberx637.1.1.m1.2c">-\frac{1}{12},\frac{1}{2}</annotation></semantics></math>’ Ground Truth: ’<math id="lstnumberx637.2.2.m2.2" class="ltx_Math" alttext="\frac{1}{2},-\frac{1}{12}" display="inline"><semantics id="lstnumberx637.2.2.m2.2a"><mrow id="lstnumberx637.2.2.m2.2.2.1" xref="lstnumberx637.2.2.m2.2.2.2.cmml"><mfrac id="lstnumberx637.2.2.m2.1.1" xref="lstnumberx637.2.2.m2.1.1.cmml"><mn mathbackground="#F47874" id="lstnumberx637.2.2.m2.1.1.2" xref="lstnumberx637.2.2.m2.1.1.2.cmml">1</mn><mn mathbackground="#F47874" id="lstnumberx637.2.2.m2.1.1.3" xref="lstnumberx637.2.2.m2.1.1.3.cmml">2</mn></mfrac><mo mathbackground="#F47874" id="lstnumberx637.2.2.m2.2.2.1.2" xref="lstnumberx637.2.2.m2.2.2.2.cmml">,</mo><mrow id="lstnumberx637.2.2.m2.2.2.1.1" xref="lstnumberx637.2.2.m2.2.2.1.1.cmml"><mo mathbackground="#F47874" id="lstnumberx637.2.2.m2.2.2.1.1a" xref="lstnumberx637.2.2.m2.2.2.1.1.cmml">−</mo><mfrac id="lstnumberx637.2.2.m2.2.2.1.1.2" xref="lstnumberx637.2.2.m2.2.2.1.1.2.cmml"><mn mathbackground="#F47874" id="lstnumberx637.2.2.m2.2.2.1.1.2.2" xref="lstnumberx637.2.2.m2.2.2.1.1.2.2.cmml">1</mn><mn mathbackground="#F47874" id="lstnumberx637.2.2.m2.2.2.1.1.2.3" xref="lstnumberx637.2.2.m2.2.2.1.1.2.3.cmml">12</mn></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="lstnumberx637.2.2.m2.2b"><list id="lstnumberx637.2.2.m2.2.2.2.cmml" xref="lstnumberx637.2.2.m2.2.2.1"><apply id="lstnumberx637.2.2.m2.1.1.cmml" xref="lstnumberx637.2.2.m2.1.1"><divide id="lstnumberx637.2.2.m2.1.1.1.cmml" xref="lstnumberx637.2.2.m2.1.1"></divide><cn type="integer" id="lstnumberx637.2.2.m2.1.1.2.cmml" xref="lstnumberx637.2.2.m2.1.1.2">1</cn><cn type="integer" id="lstnumberx637.2.2.m2.1.1.3.cmml" xref="lstnumberx637.2.2.m2.1.1.3">2</cn></apply><apply id="lstnumberx637.2.2.m2.2.2.1.1.cmml" xref="lstnumberx637.2.2.m2.2.2.1.1"><minus id="lstnumberx637.2.2.m2.2.2.1.1.1.cmml" xref="lstnumberx637.2.2.m2.2.2.1.1"></minus><apply id="lstnumberx637.2.2.m2.2.2.1.1.2.cmml" xref="lstnumberx637.2.2.m2.2.2.1.1.2"><divide id="lstnumberx637.2.2.m2.2.2.1.1.2.1.cmml" xref="lstnumberx637.2.2.m2.2.2.1.1.2"></divide><cn type="integer" id="lstnumberx637.2.2.m2.2.2.1.1.2.2.cmml" xref="lstnumberx637.2.2.m2.2.2.1.1.2.2">1</cn><cn type="integer" id="lstnumberx637.2.2.m2.2.2.1.1.2.3.cmml" xref="lstnumberx637.2.2.m2.2.2.1.1.2.3">12</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="lstnumberx637.2.2.m2.2c">\frac{1}{2},-\frac{1}{12}</annotation></semantics></math>’ Score: [False]</span>
</div>
<div id="lstnumberx638" class="ltx_listingline">
<span id="lstnumberx638.1" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Reference</span><span id="lstnumberx638.2" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.3" class="ltx_text ltx_lst_identifier ltx_font_typewriter">CoT</span><span id="lstnumberx638.4" class="ltx_text ltx_font_typewriter">:</span><span id="lstnumberx638.5" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.6" class="ltx_text ltx_font_typewriter">’</span><span id="lstnumberx638.7" class="ltx_text ltx_lst_identifier ltx_font_typewriter">We</span><span id="lstnumberx638.8" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.9" class="ltx_text ltx_lst_identifier ltx_font_typewriter">can</span><span id="lstnumberx638.10" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.11" class="ltx_text ltx_lst_identifier ltx_font_typewriter">expand</span><span id="lstnumberx638.12" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.13" class="ltx_text ltx_lst_identifier ltx_font_typewriter">using</span><span id="lstnumberx638.14" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.15" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx638.16" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.17" class="ltx_text ltx_lst_identifier ltx_font_typewriter">pairs</span><span id="lstnumberx638.18" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.19" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx638.20" class="ltx_text ltx_font_typewriter">(3</span><span id="lstnumberx638.21" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.22" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.23" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx638.24" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.25" class="ltx_text ltx_font_typewriter">1)(12</span><span id="lstnumberx638.26" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.27" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.28" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx638.29" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.30" class="ltx_text ltx_font_typewriter">1)</span><span id="lstnumberx638.31" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.32" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx638.33" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.34" class="ltx_text ltx_font_typewriter">36</span><span id="lstnumberx638.35" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.36" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx638.37" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.38" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx638.39" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.40" class="ltx_text ltx_font_typewriter">15</span><span id="lstnumberx638.41" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.42" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.43" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx638.44" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.45" class="ltx_text ltx_font_typewriter">1</span><span id="lstnumberx638.46" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx638.47" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.48" class="ltx_text ltx_lst_identifier ltx_font_typewriter">and</span><span id="lstnumberx638.49" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.50" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx638.51" class="ltx_text ltx_font_typewriter">(6</span><span id="lstnumberx638.52" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.53" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.54" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx638.55" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.56" class="ltx_text ltx_font_typewriter">1)(4</span><span id="lstnumberx638.57" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.58" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.59" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx638.60" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.61" class="ltx_text ltx_font_typewriter">1)</span><span id="lstnumberx638.62" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.63" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx638.64" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.65" class="ltx_text ltx_font_typewriter">24</span><span id="lstnumberx638.66" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.67" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx638.68" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.69" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx638.70" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.71" class="ltx_text ltx_font_typewriter">10</span><span id="lstnumberx638.72" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.73" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.74" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx638.75" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.76" class="ltx_text ltx_font_typewriter">1,</span><span id="lstnumberx638.77" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx638.78" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.79" class="ltx_text ltx_lst_identifier ltx_font_typewriter">so</span><span id="lstnumberx638.80" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx638.81" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx638.82" class="ltx_text ltx_font_typewriter">\\[(36</span><span id="lstnumberx638.83" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.84" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx638.85" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.86" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx638.87" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.88" class="ltx_text ltx_font_typewriter">15</span><span id="lstnumberx638.89" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.90" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.91" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx638.92" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.93" class="ltx_text ltx_font_typewriter">1)(24</span><span id="lstnumberx638.94" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.95" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx638.96" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.97" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx638.98" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.99" class="ltx_text ltx_font_typewriter">10</span><span id="lstnumberx638.100" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.101" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.102" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx638.103" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.104" class="ltx_text ltx_font_typewriter">1)</span><span id="lstnumberx638.105" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.106" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx638.107" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.108" class="ltx_text ltx_font_typewriter">5.\\]</span><span id="lstnumberx638.109" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Let</span><span id="lstnumberx638.110" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.111" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx638.112" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.113" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx638.114" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.115" class="ltx_text ltx_font_typewriter">12</span><span id="lstnumberx638.116" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.117" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx638.118" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.119" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx638.120" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.121" class="ltx_text ltx_font_typewriter">5</span><span id="lstnumberx638.122" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.123" class="ltx_text ltx_font_typewriter">.</span><span id="lstnumberx638.124" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx638.125" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx638.126" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Then</span><span id="lstnumberx638.127" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx638.128" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx638.129" class="ltx_text ltx_font_typewriter">\\[(3</span><span id="lstnumberx638.130" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx638.131" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.132" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx638.133" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.134" class="ltx_text ltx_font_typewriter">1)(2</span><span id="lstnumberx638.135" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx638.136" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.137" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx638.138" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.139" class="ltx_text ltx_font_typewriter">1)</span><span id="lstnumberx638.140" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.141" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx638.142" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.143" class="ltx_text ltx_font_typewriter">5.\\]</span><span id="lstnumberx638.144" class="ltx_text ltx_lst_identifier ltx_font_typewriter">This</span><span id="lstnumberx638.145" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.146" class="ltx_text ltx_lst_identifier ltx_font_typewriter">simplifies</span><span id="lstnumberx638.147" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.148" class="ltx_text ltx_lst_identifier ltx_font_typewriter">to</span><span id="lstnumberx638.149" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.150" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$6y</span><span id="lstnumberx638.151" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx638.152" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.153" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx638.154" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.155" class="ltx_text ltx_font_typewriter">5</span><span id="lstnumberx638.156" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx638.157" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.158" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx638.159" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.160" class="ltx_text ltx_font_typewriter">4</span><span id="lstnumberx638.161" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.162" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx638.163" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.164" class="ltx_text ltx_font_typewriter">0,</span><span id="lstnumberx638.165" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx638.166" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.167" class="ltx_text ltx_lst_identifier ltx_font_typewriter">which</span><span id="lstnumberx638.168" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.169" class="ltx_text ltx_lst_identifier ltx_font_typewriter">factors</span><span id="lstnumberx638.170" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.171" class="ltx_text ltx_lst_identifier ltx_font_typewriter">as</span><span id="lstnumberx638.172" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.173" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx638.174" class="ltx_text ltx_font_typewriter">(2</span><span id="lstnumberx638.175" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx638.176" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.177" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx638.178" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.179" class="ltx_text ltx_font_typewriter">1)(3</span><span id="lstnumberx638.180" class="ltx_text ltx_lst_identifier ltx_font_typewriter">y</span><span id="lstnumberx638.181" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.182" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx638.183" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.184" class="ltx_text ltx_font_typewriter">4)</span><span id="lstnumberx638.185" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.186" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx638.187" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.188" class="ltx_text ltx_font_typewriter">0.</span><span id="lstnumberx638.189" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx638.190" class="ltx_text ltx_lst_space ltx_font_typewriter">  </span><span id="lstnumberx638.191" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Hence</span><span id="lstnumberx638.192" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx638.193" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.194" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx638.195" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.196" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx638.197" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.198" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx638.199" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx638.200" class="ltx_text ltx_font_typewriter">{1}{2}</span><span id="lstnumberx638.201" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx638.202" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.203" class="ltx_text ltx_lst_identifier ltx_font_typewriter">or</span><span id="lstnumberx638.204" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.205" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$y</span><span id="lstnumberx638.206" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.207" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx638.208" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.209" class="ltx_text ltx_font_typewriter">-\\</span><span id="lstnumberx638.210" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx638.211" class="ltx_text ltx_font_typewriter">{4}{3}.</span><span id="lstnumberx638.212" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx638.213" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx638.214" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx638.215" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx638.216" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nIf</span><span id="lstnumberx638.217" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.218" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$12x</span><span id="lstnumberx638.219" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx638.220" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.221" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx638.222" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.223" class="ltx_text ltx_font_typewriter">5</span><span id="lstnumberx638.224" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.225" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.226" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx638.227" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.228" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx638.229" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx638.230" class="ltx_text ltx_font_typewriter">{1}{2},</span><span id="lstnumberx638.231" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx638.232" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.233" class="ltx_text ltx_lst_identifier ltx_font_typewriter">then</span><span id="lstnumberx638.234" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.235" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$24x</span><span id="lstnumberx638.236" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx638.237" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.238" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx638.239" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.240" class="ltx_text ltx_font_typewriter">10</span><span id="lstnumberx638.241" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.242" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.243" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx638.244" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.245" class="ltx_text ltx_font_typewriter">1</span><span id="lstnumberx638.246" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.247" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx638.248" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.249" class="ltx_text ltx_font_typewriter">0,</span><span id="lstnumberx638.250" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx638.251" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.252" class="ltx_text ltx_lst_identifier ltx_font_typewriter">which</span><span id="lstnumberx638.253" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.254" class="ltx_text ltx_lst_identifier ltx_font_typewriter">factors</span><span id="lstnumberx638.255" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.256" class="ltx_text ltx_lst_identifier ltx_font_typewriter">as</span><span id="lstnumberx638.257" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx638.258" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx638.259" class="ltx_text ltx_font_typewriter">\\[(2</span><span id="lstnumberx638.260" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.261" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.262" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx638.263" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.264" class="ltx_text ltx_font_typewriter">1)(12</span><span id="lstnumberx638.265" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.266" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.267" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx638.268" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.269" class="ltx_text ltx_font_typewriter">1)</span><span id="lstnumberx638.270" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.271" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx638.272" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.273" class="ltx_text ltx_font_typewriter">0.\\]</span><span id="lstnumberx638.274" class="ltx_text ltx_lst_identifier ltx_font_typewriter">Hence</span><span id="lstnumberx638.275" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx638.276" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.277" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x</span><span id="lstnumberx638.278" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.279" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx638.280" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.281" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx638.282" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx638.283" class="ltx_text ltx_font_typewriter">{1}{2}</span><span id="lstnumberx638.284" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx638.285" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.286" class="ltx_text ltx_lst_identifier ltx_font_typewriter">or</span><span id="lstnumberx638.287" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.288" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$x</span><span id="lstnumberx638.289" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.290" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx638.291" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.292" class="ltx_text ltx_font_typewriter">-\\</span><span id="lstnumberx638.293" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx638.294" class="ltx_text ltx_font_typewriter">{1}{12}.</span><span id="lstnumberx638.295" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx638.296" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx638.297" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx638.298" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx638.299" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nIf</span><span id="lstnumberx638.300" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.301" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$12x</span><span id="lstnumberx638.302" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx638.303" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.304" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx638.305" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.306" class="ltx_text ltx_font_typewriter">5</span><span id="lstnumberx638.307" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.308" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.309" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx638.310" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.311" class="ltx_text ltx_font_typewriter">-\\</span><span id="lstnumberx638.312" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx638.313" class="ltx_text ltx_font_typewriter">{4}{3},</span><span id="lstnumberx638.314" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx638.315" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.316" class="ltx_text ltx_lst_identifier ltx_font_typewriter">then</span><span id="lstnumberx638.317" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx638.318" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx638.319" class="ltx_text ltx_font_typewriter">\\[36</span><span id="lstnumberx638.320" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.321" class="ltx_text ltx_font_typewriter">^2</span><span id="lstnumberx638.322" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.323" class="ltx_text ltx_font_typewriter">-</span><span id="lstnumberx638.324" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.325" class="ltx_text ltx_font_typewriter">15</span><span id="lstnumberx638.326" class="ltx_text ltx_lst_identifier ltx_font_typewriter">x</span><span id="lstnumberx638.327" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.328" class="ltx_text ltx_font_typewriter">+</span><span id="lstnumberx638.329" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.330" class="ltx_text ltx_font_typewriter">4</span><span id="lstnumberx638.331" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.332" class="ltx_text ltx_font_typewriter">=</span><span id="lstnumberx638.333" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.334" class="ltx_text ltx_font_typewriter">0,\\]</span><span id="lstnumberx638.335" class="ltx_text ltx_lst_identifier ltx_font_typewriter">which</span><span id="lstnumberx638.336" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.337" class="ltx_text ltx_lst_identifier ltx_font_typewriter">has</span><span id="lstnumberx638.338" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.339" class="ltx_text ltx_lst_identifier ltx_font_typewriter">no</span><span id="lstnumberx638.340" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.341" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx638.342" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.343" class="ltx_text ltx_lst_identifier ltx_font_typewriter">solutions</span><span id="lstnumberx638.344" class="ltx_text ltx_font_typewriter">.\</span><span id="lstnumberx638.345" class="ltx_text ltx_lst_identifier ltx_font_typewriter">n</span><span id="lstnumberx638.346" class="ltx_text ltx_font_typewriter">\</span><span id="lstnumberx638.347" class="ltx_text ltx_lst_identifier ltx_font_typewriter">nTherefore</span><span id="lstnumberx638.348" class="ltx_text ltx_font_typewriter">,</span><span id="lstnumberx638.349" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.350" class="ltx_text ltx_lst_identifier ltx_font_typewriter">the</span><span id="lstnumberx638.351" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.352" class="ltx_text ltx_lst_identifier ltx_font_typewriter">real</span><span id="lstnumberx638.353" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.354" class="ltx_text ltx_lst_identifier ltx_font_typewriter">roots</span><span id="lstnumberx638.355" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.356" class="ltx_text ltx_lst_identifier ltx_font_typewriter">are</span><span id="lstnumberx638.357" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.358" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx638.359" class="ltx_text ltx_font_typewriter">\\</span><span id="lstnumberx638.360" class="ltx_text ltx_lst_identifier ltx_font_typewriter">boxed</span><span id="lstnumberx638.361" class="ltx_text ltx_font_typewriter">{\\</span><span id="lstnumberx638.362" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx638.363" class="ltx_text ltx_font_typewriter">{1}{2},</span><span id="lstnumberx638.364" class="ltx_text ltx_lst_space ltx_font_typewriter"> </span><span id="lstnumberx638.365" class="ltx_text ltx_font_typewriter">-\\</span><span id="lstnumberx638.366" class="ltx_text ltx_lst_identifier ltx_font_typewriter">frac</span><span id="lstnumberx638.367" class="ltx_text ltx_font_typewriter">{1}{12}}.</span><span id="lstnumberx638.368" class="ltx_text ltx_lst_identifier ltx_font_typewriter">$</span><span id="lstnumberx638.369" class="ltx_text ltx_font_typewriter">’</span>
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2309.17452
Large language models have made significant progress in various language tasks, yet they still struggle with complex mathematics.
In this paper, we proposeToRA, a series ofTool-integratedReasoningAgents designed to solve challenging mathematical problems by seamlessly integrating natural language reasoning with the utilization of external tools (e.g., computation libraries and symbolic solvers) , thereby amalgamating the analytical prowess of language and the computational efficiency of tools.
To trainToRA, we curate interactive tool-use trajectories on mathematical datasets, apply imitation learning on the annotations, and propose output space shaping to further refine models’ reasoning behavior.
As a result,ToRAmodels significantly outperform open-source models on 10 mathematical reasoning datasets across all scales with 13%-19% absolute improvements on average.
Notably,ToRA-7B reaches 44.6% on the competition-level dataset MATH, surpassing the best open-source model WizardMath-70B by 22% absolute.ToRA-Code-34B is also the first open-source model that achieves an accuracy exceeding 50% on MATH, which significantly outperforms GPT-4’s CoT result, and is competitive with GPT-4 solving problems with programs.
Additionally, we conduct a comprehensive analysis of the benefits and remaining challenges of tool interaction for mathematical reasoning, providing valuable insights for future research111Code and models are available athttps://github.com/microsoft/ToRA..
abstract
\logoToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving
Abstract
2309.17452
1IntroductionLarge language models (LLMs) , such as GPT-4and PaLM-2, have demonstrated remarkable progress in a wide range of language tasks, particularly in the longstanding challenge of mathematical reasoning. However, open-source models, such as LLaMA-2and Falcon, still struggle with advanced mathematical reasoning tasks.Existing works improve mathematical performance of language models either with step-by-step natural language reasoningas illustrated in Fig2(a) , or by synthesizing and executing programs to obtain the answers, as depicted in Fig2(b) .
Both approaches exhibit complementary advantages.
Natural language is suitable for semantic analysis, planning, and abstract reasoning (e.g., commonsense reasoning) , but struggles with precise computation, symbolic manipulation, and algorithmic processing.
Conversely, programs excel in rigorous operations, and can outsource intricate calculations to specialized tools like equation solvers.To leverage the benefits of both natural language reasoning and program-based tool use, we train open-source models such as LLaMA-2 to reason in a way where natural language reasoning is interleaved with program-based tool use synergistically (as depicted in Fig2(c) ) , thereby largely reducing the gap with closed-source models like GPT-4 in mathematical reasoning.
Specifically, we first design the interleaving format of reasoning, curate corresponding interactive tool-use trajectories for mathematical problems from the popular GSM8kand MATHdataset, and then apply imitation learning on the high-quality annotations, leading to a better performance than any existing open-source model.
Furthermore, since the curated data is far from exhausting all valid trajectories for a problem, relying solely on imitation learning restricts a model’s output space, hindering the flexibility in exploring plausible trajectories during testing.
To improve the diversity of plausible reasoning steps and mitigate improper tool-use behavior, we applyoutput space shapingwhich additionally trains the models on both self-sampled valid trajectories and invalid ones that have been corrected by a teacher model (e.g., a 34B model can serve as the teacher for a 7B model) .Output space shapingsignificantly boosts reasoning, allowing open-source models to attain an accuracy exceeding 50% on the competition-level MATH dataset for the first time.We evaluate the resulting suite ofTool-integratedReasoningAgents (ToRA)  ranging from 7B to 70B on 10 diverse mathematical reasoning datasets.
As shown in Fig1,ToRAseries significantly outperform open-source models across all scales.
Notably, on the competition-level MATH dataset,ToRA-7B outperforms the previous SoTA WizardMath-70Bby 22% absolute.ToRA-Code-34B beats GPT-4’s CoT resultby 8.3% absolute (50.8% vs. 42.5%) , and is competitive with GPT-4 solving problems with code (GPT-4-Code, 51.8%) .
In addition, we analyze the benefits and remaining challenges of tool interaction for mathematical reasoning, providing valuable insights for future work.
introduction
\logoToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving
1Introduction
2309.17452
2ToRA: Tool-Integrated Agents for Mathematical ReasoningFigure 3:TrainingToRAcontains two steps.
①Imitation Learning: Prompt LLMs like GPT-4 to generate Tool-integrated Reasoning trajectories (ToRA-Corpus)  and use this corpus to fine-tune a model;
②Output Space Shaping: Sample diverse tool-use trajectories with, keep the valid ones, correct the invalid ones with a teacher model, and retrainon the union of sampled valid trajectories, corrected ones, and the initialToRA-Corpusto obtainToRA.2.1OverviewToRAseries solve challenging mathematical problems by leveraging both natural language reasoning and program-based tool use.
As shown in Fig2(c) , given a mathematical problem,ToRAreasons with natural language, producing.
When reaching a point where program-based tool use is more appropriate for the subsequent task, e.g., equation solving,ToRAgenerates a programfor tool use following natural language guidance.
The execution outputwill be fed toToRAfor subsequent processing including tool use adjustments, sub-tasks solving, or answer finalization.
We repeat the process until the model places its answer within “\boxed{}”.
The resulting trajectory is denoted as, wherecontains the answer.Fig3presents the training pipeline ofToRA.
We first collect interactive tool-use trajectories on popular mathematical datasets.
We then apply imitation learning on the resulting annotations, as well as output space shaping to further refine models’ reasoning behavior.2.2Collecting Interactive Tool-Use Trajectories1:problem, model, prompt, external tools, stop conditionStop() , maximum iteration rounds2:Trajectory Initialization3:fortodo4:Rationale Generation (Eq.1) 5:ifStop() thenStopping Criteria6:return7:endif8:Program Generation (Eq.2) 9:Tool Execution10:Trajectory Update (Eq.3) 11:endfor12:returnAlgorithm 1Inference of Tool-Integrated ReasoningExisting mathematical reasoning datasets primarily contain annotations in either natural language or code, posing a challenge for training tool-integrated agents due to the absence of interactive tool-use annotations.
To address this, we utilize GPT-4 to synthesize high-quality trajectories on the GSM8k and MATH training sets. We select GSM8k and MATH as they exhibit diverse reasoning patterns, spanning multiple domains and difficulty levels.Prompt CurationWe compose instructions along with diverse few-shot examples, utilizing an interleaved format as depicted in Fig2(c) . These examples showcase interactive tool usage trajectories, incorporating descriptive variable names and combined program outputs. Please refer to AppendixEfor the assembled prompts.Inference ProcedureWe follow Algorithm1and feed GPT-4 ()  with the composed promptto generate a tool-use trajectoryfor each questionfrom the training set.
The trajectory is initialized as an empty string, for each interaction round, we first generate a rationale:(1) wheremeans concatenation. Ifincludes an answer within “\boxed{}” (i.e., the stopping conditionStop() ) , we cease generation, otherwise the model continues to write a program for tool use:(2) In line with, if the model triggers the code execution stop words like “‘‘‘output”, we supply it with the corresponding execution message and outputby calling tools with, facilitating the generation of subsequent steps. Then, we update the trajectory by concatenating it with the newly generated rationale, program, and output:(3) We repeat the above interaction process until we reach the maximum rounds.Trajectory SamplingWe setand perform inference using GPT-4 with greedy decoding, retaining trajectories that yield correct answers. For questions where GPT-4 fails with greedy decoding, we apply nucleus sampling with a sample size of 10 and keep up to 4 valid trajectories per question.
Ultimately, we successfully annotate trajectories for 98.2% of GSM8k questions and 83.1% of MATH questions.
After filtering out invalid trajectories with tool-use errors or wrong answers, we obtain 16k annotations which constitute our datasetToRA-Corpus.
Table1comparesToRA-Corpuswith recently proposed mathematical reasoning datasets, while Table6in the Appendix displays MATH annotation accuracy details.Table 1:Compared with mathematical reasoning datasets,ToRA-Corpusuniquely combines natural language rationales with program-based tool usage.
Note thatToRA-Corpusonly employ questions from the original training set of MATH and GSM8k.Methods#AnnotationToolInterleavingLLM UsedSourceRFT100k✗✗LLaMA-2GSM8kOpen-Platypus25k✗✗GPT-411 datasets with MATHWizardMath96k✗✗ChatGPTMATH & GSM8kLila134k✓(PoT) ✗-20 datasets with MATH & GSM8kMathInstruct260k✓(PoT) ✗GPT-414 datasets with MATH & GSM8kToRA-Corpus(ours) 16k✓✓GPT-4MATH & GSM8k2.3TrainingImitation LearningWe apply imitation learning onToRA-Corpusby minimizing negative log-likelihood loss on the trajectoryconditioned on the problem:(4) whereis the resulting model.
After imitation learning, we can simply apply the same procedure in Algorithm1by setting prompt to emptyfor inference.
Imitation learning leads to state-of-the-art mathematical reasoning performance despite the small scale ofToRA-Corpus.Output Space ShapingFor each question,ToRA-Corpusmostly demonstrates only one valid interactive tool-use trajectory, which may restrict a model’s output space, rendering it inflexible in exploring plausible trajectories during testing.
We therefore proposeoutput space shapingin order to encourage the diversity of plausible reasoning steps and reduce improper tool-use behavior.To explore diverse valid trajectories, we apply nucleus sampling to imitation learning modelsto sample 64 trajectories per training question, following the inference procedure in Section2.2.
We retain valid trajectories with correct answers and no tool-use errors.
As many samples are duplicates, to further improve diversity and in an attempt to correct models’ improper behavior, we seek to leverage invalid trajectories as well.
We observe that trajectories with wrong answers are mostly incorrect halfway, and the preceding reasoning is still plausible; in other words, we can obtain valid trajectories by correcting the subsequent portions.
Specifically, a wrong trajectory, when written in text, can be represented as a sequence of lines separated by line breaks, i.e.,, whereis the total number of lines in.
We enumerate possible preceding portions of wrong trajectories, i.e.,, and leverage a teacher modelto complete the subsequent steps with greedy decoding:where we abuse the notationto denote the interactive tool use process following Section2.2.
Finally, corrected trajectories as well as valid trajectory samples will be used for model training, thereby shaping the output space.In our experiments, we always use CodeLLaMA-34B trained onToRA-Corpusas the teacher model, and apply sampling with the CodeLLaMA series (ranging from 7B to 34B, with imitation learning) .
We obtain a total of 233k distinct valid trajectory samples and 69k corrected ones.
From this combined dataset, we randomly select up to 4 trajectories per GSM8k and MATH problem, merge them withToRA-Corpus, and then train allToRAmodels on the resulting 69k annotations.
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\logoToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving
2ToRA: Tool-Integrated Agents for Mathematical Reasoning
2309.17452
3Experiments3.1Implementation DetailsWe fine-tuned LLaMA-2and CodeLLaMAseries (ranging from 7B to 70B)  usingToRA-Corpuswith output space shaping, yielding theToRAandToRA-Codeseries respectively.
We used a learning rate of 2e-5 by default except that we used 1e-5 for the 34B and 70B models.
We set the global batch size to 128 and used a linear scheduler with a 3% warm-up period for 3 epochs.
We trained all models withDeepSpeed ZeRO Stage3andFlash-Attention 2.
We used greedy decoding for all results, with the maximum sequence length set to 2,048 and the maximum number of tool executions set to 3.3.2Evaluation SetupDatasetsWe evaluated models on GSM8kand MATH, along with 8 out-of-distribution datasets, namely GSM-Hard, SVAMP, ASDIV, TabMWP, SingleEQ, SingleOP, AddSub, and MultiArith, as illustrated in Table5in Appendix.
The 10 assorted datasets collectively encompass mathematical problems spanning basic arithmetic to competition level, covering middle and high school curricula and various mathematical domains.
The problem formats comprise tabular-based, free-form, and multiple-choice questions, ensuring a thorough assessment of the model’s mathematical reasoning aptitude.MetricsWe report accuracies of predicted answers.
Following, we round numerical values and usesympy222https://www.sympy.orgfor parsing expressions.
Since the SingleEQ, SingleOP, AddSub, and MultiArith datasets focus on different aspects of basic arithmetic, we report their average results under the collective term MAWPSfor all methods.3.3BaselinesProprietary ModelsWe present results from an array of SoTA LLMs, such as OpenAI’s GPT-4, ChatGPT (gpt-3.5-turbo) , Google’s PaLM-2, and Anthropic’s Claude-2. By default, we report CoT prompting results, and include PALprompting results for selected models.Open-Source ModelsBase modelscomprise LLaMA-2 and CodeLLaMA with CoT and PAL prompting.Supervised Fine-Tuning (SFT) employs CoT rationales from the original GSM8k and MATH dataset (15k samples)  for fine-tuning.Rejection sampling Fine-Tuning (RFT) leverages multiple models to generate diverse reasoning paths for fine-tuning.WizardMathaugments data using ChatGPT, and conducts SFT and RLHF.Platypus-2, the top model on the LLM Leaderboard333https://huggingface.co/spaces/HuggingFaceH4/open_llm_leaderboard, is fine-tuned with Open-Platypus reasoning datasets.
We also compareToRAwith Toolformerwhich is a model trained to utilize calculators.Table 2:Results on 10 mathematical reasoning tasks. MAWPS results are averaged over four tasks: Singleeq, Singleop, Addsub, and MultArith. Vanilla models are tested with CoT. The best results in each section are inblue, the second-best results are underlined, while the results of our best model are bolded.∗ZS: Zero-shot inference without demonstrations.ModelSizeToolsZS∗GSM8kMATHGSM-HardSVAMPTabMWPASDivMAWPSAVGUsed for training?✓✓✗✗✗✗✗Proprietary ModelsGPT-4-✗✗92.042.564.793.167.191.397.678.3GPT-4 (PAL) -✓✗94.251.877.694.895.992.697.786.4ChatGPT-✗✗80.835.555.983.069.187.394.672.3ChatGPT (PAL) -✓✗78.638.767.677.879.981.089.473.3Claude-2-✗✗85.232.5------PaLM-2540B✗✗80.734.3------Open-Source ModelsLLaMA-27B✗✗13.34.17.838.031.150.760.929.4LLaMA-2 SFT7B✗✓41.37.216.131.927.847.460.033.1LLaMA-2 RFT7B✗✓51.2-------Platypus-27B✗✗14.45.48.636.726.547.958.428.3WizardMath7B✗✓54.910.720.657.338.159.173.744.9CodeLLaMA (PAL) 7B✓✗34.016.633.659.047.361.479.647.4Toolformer†7B✓✓---29.4-40.444.0-ToRA7B✓✓68.840.154.668.242.473.988.862.4ToRA-Code7B✓✓72.644.656.070.451.678.791.366.5 (+19) LLaMA-213B✗✗24.36.313.643.139.556.370.436.2LLaMA-2 SFT13B✗✓51.19.222.346.335.858.675.042.6LLaMA-2 RFT13B✗✓55.3-------Platypus-213B✗✗23.77.114.350.745.355.169.638.0WizardMath13B✗✓63.914.028.464.346.765.879.751.8CodeLLaMA (PAL) 13B✓✗39.919.939.062.459.565.386.053.1ToRA13B✓✓72.743.057.372.947.277.291.365.9ToRA-Code13B✓✓75.848.160.575.765.481.492.571.3 (+18) LLaMA-1 RFT34B✗✓57.9-------CodeLLaMA (PAL) 34B✓✗53.323.949.471.063.172.491.560.7ToRA-Code34B✓✓80.750.863.780.570.584.293.374.8 (+14) LLaMA-270B✗✗57.814.436.073.657.576.092.458.2LLaMA-2 SFT70B✗✓69.314.939.064.053.071.384.856.6LLaMA-2 RFT70B✗✓64.8-------Platypus-270B✗✗45.915.024.674.347.372.791.153.0WizardMath70B✗✓81.622.750.380.049.876.286.263.8LLaMA-2 (PAL) 70B✓✗55.218.350.074.659.571.992.860.3ToRA70B✓✓84.349.767.282.774.086.893.876.9 (+13) 3.4Main ResultsTable 3:Results on MATH subtopics.ModelSizeToolIntermediateAlgebraPrecalculusGeometryNumberTheoryCounting&ProbabilityPrealgebraAlgebraOverallProprietary ModelsChatGPT (PAL) -✓18.519.223.248.543.062.745.438.7GPT-4 (PAL) -✓32.829.338.058.761.073.959.151.8Open-Source ModelsWizardMath7B✗6.26.06.57.69.518.116.311.2ToRA-Code7B✓35.1 (+28.9) 31.0 (+25.0) 24.0 (+17.5) 50.7 (+43.1) 30.6 (+21.1) 55.0 (+36.9) 61.7 (+45.4) 44.6 (+33.4) w/o Shaping7B✓29.7 (-5.4) 25.1 (-5.9) 17.7 (-6.3) 46.9 (-3.8) 32.3 (+1.7) 51.9 (-3.1) 55.7 (-6.0) 40.2 (-4.4) w/o Rationale7B✓25.5 (-9.6) 14.7 (-16.3) 15.4 (-8.6) 45.9 (-4.8) 29.7 (-0.9) 51.0 (-4.0) 52.4 (-9.3) 36.8 (-7.8) WizardMath13B✗6.46.611.59.611.028.521.115.0ToRA-Code13B✓35.7 (+29.3) 31.1 (+24.5) 25.7 (+14.2) 55.6 (+46.0) 39.5 (+28.5) 58.7 (+30.2) 66.7 (+45.6) 48.1 (+33.1) w/o Shaping13B✓32.8 (-2.9) 26.0 (-5.1) 24.0 (-1.7) 52.6 (-3.0) 38.4 (-1.1) 55.6 (-3.1) 61.2 (-5.5) 44.6 (-3.5) w/o Rationale13B✓27.1 (-8.6) 15.8 (-15.3) 16.3 (-9.4) 50.4 (-5.2) 36.9 (-2.6) 55.3 (-3.4) 56.5 (-10.2) 40.2 (-7.9) ToRA-Code34B✓38.934.627.357.841.463.767.750.8w/o Shaping34B✓34.0 (-4.9) 29.9 (-4.7) 24.6 (-2.7) 55.6 (-2.2) 41.6 (+0.2) 63.8 (+0.1) 61.4 (-6.3) 47.4 (-3.4) w/o Rationale34B✓28.3 (-10.6) 15.8 (-18.8) 18.0 (-9.3) 52.4 (-5.4) 40.7 (-0.7) 58.6 (-5.1) 57.5 (-10.2) 41.9 (-8.9) WizardMath70B✗9.113.416.916.519.242.735.024.1ToRA70B✓37.1 (+28) 30.4 (+17) 30.1 (+13.2) 54.6 (+38.1) 40.3 (+21.1) 64.9 (+22.2) 66.6 (+31.6) 49.7 (+25.6) w/o Shaping70B✓33.8(-3.3) 28.9(-1.5) 27.1(-3) 53.0(-1.6) 38.0(-2.3) 62.2(-2.7) 64.2(-2.4) 47.3(-2.4) w/o Rationale70B✓26.7(-10.4) 14.7(-15.7) 20.3(-9.8) 48.9(-5.7) 39.2(-1.1) 59.8(-5.1) 57.6(-9) 41.5(-8.2) Table2presents the results ofToRAon 10 mathematical datasets, highlighting the following salient observations:(1) Using interleaved formatting and output space shaping,ToRAconsistently surpasses prior state-of-the-art open-source models across all scales, achieving 13% to 19% absolute improvements across 10 tasks.(2) ToRA-70B substantially outperforms ChatGPT with both CoT and PAL prompting on GSM8k (84.3% vs. 80.4%)  and MATH (49.7% vs. 38.7%) , whileToRA-Code-34B is competitive with GPT-4 solving competition-level MATH dataset with code (50.8% vs. 51.8%) .(3) The accuracy ofToRA-Codeis about 5% higher thanToRAof the same size, demonstrating that continued training on code data significantly benefits program-based tool use.(4) While rationale-based fine-tuning negatively affects out-of-distribution generalization,ToRAdisplays superior generalization. For instance, WizardMath-70B underperforms the base model on TabMWP (49.8% vs. 57.5%) , whileToRA-70B effectively generalizes to this tabular reasoning task (74.0%) .(5) ToRAattains fast zero-shot inference speed, averaging 1.02 tool interaction rounds per problem, while effectively addressing problems that require interactive tool utilization.3.5Ablation StudyFigure 4:Comparison of three formats:
(1)  Rationale-only: step-by-step natural language reasoning like CoT;
(2)  Program-only: solving problems with programs like PAL;
(3)  Tool-integrated Reasoning used byToRA: interweaving rationale and program execution to solve problems.
We evaluated GPT-4 with few-shot prompting.
We trained LLaMA-2 models to reason in the three types of formats, respectively.
For a fair comparison, wedo not apply output space shapingfor all LLaMA-2 models.3.5.1Comparisons of FormattingTo evaluate the efficacy of the reasoning format adopted byToRAwhich interleaves rationales with programs, we compared it with Rationale-only and Program-only formats using GPT-4 and LLaMA-2 trained with the same size of data from MATH.
As shown in Fig4, theToRAmethod consistently surpasses Rationale-only and Program-only approaches.
Remarkably, using LLaMA-2, theToRAmethod achieves substantial improvements of 29.0% and 6.7% over Rationale-only and Program-only, respectively.
With the closed-source GPT-4, the improvements are 19.1% and 9.8%, respectively.
This emphasizes the effectiveness of integrating natural language rationales with programs.3.5.2Effects of Output Space ShapingFigure 5:Ablation on output space shaping strategies using CodeLLaMA:
(1) ToRAis trained onToRA-Corpuswithout shaping. (2) ToRAemploys only the sampling strategy for shaping, trained with up to 4 additional valid trajectory samples per problem. (3) ToRAutilizes both the sampling and correction, also trained with up to 4 additional trajectories per problem.We assess the effectiveness of the output space shaping strategies presented in Section2.3, specifically sampling and correction.
As shown in Fig5and Table3:
(1)  Output space shaping yields a considerable average improvement of 3.4% and 4.0% absolute for GSM8k and MATH, respectively, with greater benefits for smaller models;
(2)  Applying the sampling strategy results in a 2.7% absolute improvement on average, while additionally incorporating correction offers a more substantial boost of up to 4.5%, without using more training data;
(3)  Output space shaping benefits even the largest modelToRA-70B, with a notable improvement from 47.3% to 49.7% on MATH.
These findings highlight the effectiveness of our shaping strategies across different model sizes and datasets.3.6AnalysisWe investigate the benefits, detailed patterns, and remaining challenges of tool interaction for mathematical reasoning on the challenging MATH dataset.
Performance breakdowns on all subtopics of MATH are reported in Table3.Benefits from Tool-Integration for MATH Sub-topicsAs shown in Table3,ToRAoutperforms WizardMath by around 45% in Algebra and Number Theory, which is attributed to stimulating and shaping tool-use behavior.
Problems from the two sub-topics typically need intricate computation and data manipulation.
Algebra mainly focuses on solving equations and application problems, while many Number Theory problems can be tackled using brute-force approaches through code.Figure 6:Library usage frequency and accuracy on each sub-topic of MATH.Patterns of Library Usage for Problem SolvingFig6presents the most frequently used libraries for different sub-topics and the corresponding accuracies of their solutions.
Tool-use behavior on different mathematical areas demonstrates distinct patterns.sympyand its internalsolversare primarily employed for algebra-related topics.
Precalculus exhibits extensive matrix operations viamatrices, resulting in a high accuracy.
Number Theory depends onalgorithmslikegcdandlcm.
Geometry mainly uses therationallibrary for fraction-based computations, while the application of other tools is limited, signifying the potential for improvement.Detailed Impact of Rationale on Different TopicsTable3shows that using an interleaved format, in contrast to merely writing the program, leads to significant improvements across all subtopics, especially in Precalculus, Algebra, and Geometry, where notable increases range from 8.6% to 18.8%. AppendixF.1provides representative examples demonstrating how the rationale aids in planning, multi-round self-correction, and finalizing answers.Table 4:The failure modes of theToRAon MATH, and their corresponding percentages in random samples analyzed by humans. We include specific examples of each failure mode in AppendixF.Error TypeDefinition%ExamplesReasoning ErrorMistakes due to incorrect reasoning steps or missing conditions.38%Ex.5HallucinationFabrication of numbers or answers.5%Ex.6Diagram UnderstandingMisinterpretation of the input diagram.21%Ex.7Inappropriate Tool UseIncorrect use of external tools, especially whenthe problem can’t be solved directly with libraries.10%Ex.8Syntax ErrorPersistent syntax errors despite multiple correction attempts.9%Ex.9Runtime ErrorErrors during program execution, unresolved by retrying.9%Ex.10Rationale-only ErrorCannot be formalized into a program and the rationale is incorrect.3%Ex.11False NegativeCorrect answers that don’t fully match the ground truth.5%Ex.12Remaining Challenges in Mathematical Reasoning forToRATo better understand the failure modes and remaining challenges,
we manually annotated 100 randomly selected trajectories from the MATH test set, identifying and categorizing their failure modes.
The results are shown in Table4:
Primarily, incorrect reasoning steps constitute the primary source of errors for ToRA on complex math reasoning tasks (38%) , with some hallucination issues also evident during problem interpretation and answer finalization (5%) .
Secondly, the misinterpretation of input diagrams contributes significantly to the error rate (21%) . This is particularly noticeable in Geometry, Precalculus, and Intermediate Algebra. The diagrams in the MATH dataset are usually detailed in text using the Asymptote language, thus making it challenging forToRAto comprehend diagrams purely from textual descriptions.
Thirdly, issues with tool usage include Inappropriate Tool Usage (10%) , Syntax Error (9%) , and Runtime Error (9%) .
These problems frequently arise whenToRAfails to use tools correctly after several corrections or attempts. There are certain inputs that fail to formalize well as programs (3%) , which require abstract reasoning rather than computation.
Finally, we also found that there are false negatives when using automatic indicators, i.e., correct predictions that are misjudged as wrong, but the proportion is relatively small (5%) .
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\logoToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving
3Experiments
2309.17452
4ConclusionThis paper presentsToRA, a series of novel Tool-integrated Reasoning Agents that synergistically combines natural language rationale with program-based tool-use for mathematical problem solving. Our approach demonstrates the potential of integrating external tools in the reasoning process, enabling language models to effectively tackle complex quantitative tasks.ToRAachieves state-of-the-art performance on 10 diverse mathematical reasoning tasks, substantially outperforming existing rationale-based and program-based approaches. Furthermore, our systematic analysis of the benefits and remaining challenges of tool interaction provides valuable insights for future research, contributing to the development of more advanced and versatile reasoning agents.Author ContributionsZhibin Gou proposed the interleaved tool-use format ofToRAand curatedToRA-Corpusdataset, implemented the training and evaluation pipeline, conducted experiments and analysis on all datasets, implemented baselines, and was a main contributor to the paper writing.
Zhihong Shao proposed the project, conducted preliminary experiments, proposed and implemented the training and evaluation pipelines, proposed and trained allToRAmodels with output space shaping as well asToRAvariants in the ablation study, designed and oversaw experimental analysis, and contributed to many parts of the paper writing.
Yeyun Gong, Yelong Shen, Yujiu Yang, Minlie Huang, Nan Duan, and Weizhu Chen provided research mentorship, oversaw project coordination, and advised and contributed to many parts of the writing.AcknowledgmentsZhibin Gou and Yujiu Yang were supported by the National Natural Science Foundation of China (Grant No. U1903213)  and the Shenzhen Science and Technology Program (JSGG20220831110203007) .
Zhihong Shao and Minlie Huang were supported by the NSFC projects (Key project with No. 61936010 ) , and were also supported by the National Science Foundation for Distinguished Young Scholars (with No. 62125604) .
conclusion
\logoToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving
4Conclusion