Update README.md

README.md

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 ---
-library_name: transformers
-tags: []
+license: apache-2.0
+datasets:
+- qingy2024/PR2-SFT
+language:
+- en
+base_model:
+- Qwen/Qwen2.5-14B-Instruct
 ---
 
-# Model Card for Model ID
+## PR2 14B Instruct
 
-<!-- Provide a quick summary of what the model is/does. -->
+PR2 is an experimental fine tune of Qwen2.5 14B Instruct, incorporating planning and reasoning before outputting a final answer. It is trained for 2 epochs on the verified subset of qingy2024/PR2-SFT.
 
+**Example:**
 
+Input: Find the minimum value of the expression \\(\left(\sqrt{2(1+\cos 2 x)}-\sqrt{36-4 \sqrt{5}} \sin x+2\right) \cdot \left(3+2 \sqrt{10-\sqrt{5}} \cos y-\cos 2 y\right)\\). If the answer is not an integer, round it to the nearest integer.
 
-## Model Details
+Output:
 
-### Model Description
+```
+<plan>
+1. Simplify the expression by breaking it down into manageable parts.
+2. Use trigonometric identities to simplify terms involving trigonometric functions.
+3. Analyze the structure of the expression to identify potential optimization strategies.
+4. Consider the possibility of minimizing each part of the expression separately.
+5. Calculate the minimum value of the entire expression and round it to the nearest integer if necessary.
+</plan>
 
-<!-- Provide a longer summary of what this model is. -->
+<think>
+Hmm... let me start by simplifying the expression. The expression is:
 
-This is the model card of a 🤗 transformers model that has been pushed on the Hub. This model card has been automatically generated.
+\[
+\left(\sqrt{2(1+\cos 2x)} - \sqrt{36-4\sqrt{5}} \sin x + 2\right) \cdot \left(3 + 2\sqrt{10-\sqrt{5}} \cos y - \cos 2y\right)
+\]
 
-- **Developed by:** [More Information Needed]
-- **Funded by [optional]:** [More Information Needed]
-- **Shared by [optional]:** [More Information Needed]
-- **Model type:** [More Information Needed]
-- **Language(s) (NLP):** [More Information Needed]
-- **License:** [More Information Needed]
-- **Finetuned from model [optional]:** [More Information Needed]
+First, I notice that \(\sqrt{2(1+\cos 2x)}\) can be simplified using a trigonometric identity. Recall that \(1 + \cos 2x = 2\cos^2 x\), so:
 
-### Model Sources [optional]
+\[
+\sqrt{2(1+\cos 2x)} = \sqrt{2 \cdot 2\cos^2 x} = \sqrt{4\cos^2 x} = 2|\cos x|
+\]
 
-<!-- Provide the basic links for the model. -->
+Since \(\cos x\) can be positive or negative, we'll keep it as \(2|\cos x|\) for now.
 
-- **Repository:** [More Information Needed]
-- **Paper [optional]:** [More Information Needed]
-- **Demo [optional]:** [More Information Needed]
+Next, let's look at the second part of the first factor: \(-\sqrt{36-4\sqrt{5}} \sin x + 2\). This seems a bit complicated, but perhaps we can find a way to simplify or bound it.
 
-## Uses
+Now, let's consider the second factor: \(3 + 2\sqrt{10-\sqrt{5}} \cos y - \cos 2y\). Again, we can use trigonometric identities to simplify \(\cos 2y\). Recall that \(\cos 2y = 2\cos^2 y - 1\), so:
 
-<!-- Address questions around how the model is intended to be used, including the foreseeable users of the model and those affected by the model. -->
+\[
+3 + 2\sqrt{10-\sqrt{5}} \cos y - \cos 2y = 3 + 2\sqrt{10-\sqrt{5}} \cos y - (2\cos^2 y - 1) = 4 + 2\sqrt{10-\sqrt{5}} \cos y - 2\cos^2 y
+\]
 
-### Direct Use
+This is a quadratic in terms of \(\cos y\), so we might be able to find its minimum value.
 
-<!-- This section is for the model use without fine-tuning or plugging into a larger ecosystem/app. -->
+Wait, perhaps it's better to think about minimizing each factor separately and then multiply the minima together. Let's try that.
 
-[More Information Needed]
+First, let's find the minimum of the first factor:
 
-### Downstream Use [optional]
+\[
+F_1(x) = 2|\cos x| - \sqrt{36-4\sqrt{5}} \sin x + 2
+\]
 
-<!-- This section is for the model use when fine-tuned for a task, or when plugged into a larger ecosystem/app -->
+This seems complicated due to the absolute value and the trigonometric functions. Maybe we can express it in terms of a single trigonometric function.
 
-[More Information Needed]
+Alternatively, perhaps we can use calculus to find the minimum of \(F_1(x)\). Let's try that.
 
-### Out-of-Scope Use
+Let's consider \(F_1(x) = 2|\cos x| - \sqrt{36-4\sqrt{5}} \sin x + 2\). To find the minimum, we can take the derivative and set it to zero.
 
-<!-- This section addresses misuse, malicious use, and uses that the model will not work well for. -->
+However, the absolute value makes this tricky. Perhaps we can consider the cases where \(\cos x\) is positive and negative separately.
 
-[More Information Needed]
+Case 1: \(\cos x \geq 0\)
 
-## Bias, Risks, and Limitations
+Then \(F_1(x) = 2\cos x - \sqrt{36-4\sqrt{5}} \sin x + 2\). Taking the derivative with respect to \(x\):
 
-<!-- This section is meant to convey both technical and sociotechnical limitations. -->
+\[
+F_1'(x) = -2\sin x - \sqrt{36-4\sqrt{5}} \cos x
+\]
 
-[More Information Needed]
+Setting \(F_1'(x) = 0\):
 
-### Recommendations
+\[
+-2\sin x - \sqrt{36-4\sqrt{5}} \cos x = 0 \implies -2\sin x = \sqrt{36-4\sqrt{5}} \cos x \implies \tan x = -\frac{\sqrt{36-4\sqrt{5}}}{2}
+\]
 
-<!-- This section is meant to convey recommendations with respect to the bias, risk, and technical limitations. -->
+This gives us a critical point, but we need to check if it's a minimum.
 
-Users (both direct and downstream) should be made aware of the risks, biases and limitations of the model. More information needed for further recommendations.
+Alternatively, perhaps we can use trigonometric identities to express \(F_1(x)\) as a single trigonometric function.
 
-## How to Get Started with the Model
+Recall that \(A\sin x + B\cos x = R\sin(x + \alpha)\), where \(R = \sqrt{A^2 + B^2}\) and \(\tan \alpha = \frac{B}{A}\).
 
-Use the code below to get started with the model.
+In our case, \(A = -\sqrt{36-4\sqrt{5}}\) and \(B = 2\), so:
 
-[More Information Needed]
+\[
+R = \sqrt{(-\sqrt{36-4\sqrt{5}})^2 + 2^2} = \sqrt{36-4\sqrt{5} + 4} = \sqrt{40-4\sqrt{5}} = 2\sqrt{10-\sqrt{5}}
+\]
 
-## Training Details
+So, \(F_1(x) = 2\sqrt{10-\sqrt{5}} \sin(x + \alpha) + 2\).
 
-### Training Data
+The minimum value of \(\sin(x + \alpha)\) is \(-1\), so the minimum of \(F_1(x)\) is:
 
-<!-- This should link to a Dataset Card, perhaps with a short stub of information on what the training data is all about as well as documentation related to data pre-processing or additional filtering. -->
+\[
+2\sqrt{10-\sqrt{5}} \cdot (-1) + 2 = -2\sqrt{10-\sqrt{5}} + 2
+\]
 
-[More Information Needed]
+Now, let's find the minimum of the second factor:
 
-### Training Procedure
+\[
+F_2(y) = 4 + 2\sqrt{10-\sqrt{5}} \cos y - 2\cos^2 y
+\]
 
-<!-- This relates heavily to the Technical Specifications. Content here should link to that section when it is relevant to the training procedure. -->
+This is a quadratic in \(\cos y\). Let \(z = \cos y\), then:
 
-#### Preprocessing [optional]
+\[
+F_2(y) = -2z^2 + 2\sqrt{10-\sqrt{5}} z + 4
+\]
 
-[More Information Needed]
+This is a quadratic equation in \(z\), and since the coefficient of \(z^2\) is negative, it has a maximum value. However, we need the minimum value of \(F_2(y)\). The minimum will occur at the boundaries of \(z\), i.e., when \(z = 1\) or \(z = -1\).
 
+Let's evaluate \(F_2(y)\) at \(z = 1\) and \(z = -1\).
 
-#### Training Hyperparameters
+At \(z = 1\):
 
-- **Training regime:** [More Information Needed] <!--fp32, fp16 mixed precision, bf16 mixed precision, bf16 non-mixed precision, fp16 non-mixed precision, fp8 mixed precision -->
+\[
+F_2(y) = -2(1)^2 + 2\sqrt{10-\sqrt{5}} (1) + 4 = -2 + 2\sqrt{10-\sqrt{5}} + 4 = 2 + 2\sqrt{10-\sqrt{5}}
+\]
 
-#### Speeds, Sizes, Times [optional]
+At \(z = -1\):
 
-<!-- This section provides information about throughput, start/end time, checkpoint size if relevant, etc. -->
+\[
+F_2(y) = -2(-1)^2 + 2\sqrt{10-\sqrt{5}} (-1) + 4 = -2 - 2\sqrt{10-\sqrt{5}} + 4 = 2 - 2\sqrt{10-\sqrt{5}}
+\]
 
-[More Information Needed]
+Since \(\sqrt{10-\sqrt{5}} > 0\), the minimum value of \(F_2(y)\) is \(2 - 2\sqrt{10-\sqrt{5}}\).
 
-## Evaluation
+Now, the minimum value of the entire expression is the product of the minima of \(F_1(x)\) and \(F_2(y)\):
 
-<!-- This section describes the evaluation protocols and provides the results. -->
+\[
+(-2\sqrt{10-\sqrt{5}} + 2) \cdot (2 - 2\sqrt{10-\sqrt{5}}) = (-2\sqrt{10-\sqrt{5}} + 2)(2 - 2\sqrt{10-\sqrt{5}})
+\]
 
-### Testing Data, Factors & Metrics
+Let's compute this product:
 
-#### Testing Data
+\[
+(-2\sqrt{10-\sqrt{5}} + 2)(2 - 2\sqrt{10-\sqrt{5}}) = (-2\sqrt{10-\sqrt{5}} \cdot 2) + (-2\sqrt{10-\sqrt{5}} \cdot -2\sqrt{10-\sqrt{5}}) + (2 \cdot 2) + (2 \cdot -2\sqrt{10-\sqrt{5}})
+\]
 
-<!-- This should link to a Dataset Card if possible. -->
+Simplifying each term:
 
-[More Information Needed]
+\[
+= -4\sqrt{10-\sqrt{5}} + 4(10-\sqrt{5}) + 4 - 4\sqrt{10-\sqrt{5}}
+\]
 
-#### Factors
+Combine like terms:
 
-<!-- These are the things the evaluation is disaggregating by, e.g., subpopulations or domains. -->
+\[
+= -4\sqrt{10-\sqrt{5}} - 4\sqrt{10-\sqrt{5}} + 4(10-\sqrt{5}) + 4 = -8\sqrt{10-\sqrt{5}} + 40 - 4\sqrt{5} + 4 = -8\sqrt{10-\sqrt{5}} + 44 - 4\sqrt{5}
+\]
 
-[More Information Needed]
+This expression is quite complicated, and calculating its exact value might not be straightforward. However, we can approximate the value.
 
-#### Metrics
+First, let's approximate \(\sqrt{10 - \sqrt{5}}\). Since \(\sqrt{5} \approx 2.236\), we have:
 
-<!-- These are the evaluation metrics being used, ideally with a description of why. -->
+\[
+\sqrt{10 - 2.236} = \sqrt{7.764} \approx 2.786
+\]
 
-[More Information Needed]
+Similarly, \(\sqrt{5} \approx 2.236\).
 
-### Results
+Now, let's approximate the expression:
 
-[More Information Needed]
+\[
+-8 \cdot 2.786 + 44 - 4 \cdot 2.236 = -22.288 + 44 - 8.944 = 12.768
+\]
 
-#### Summary
+Rounding this to the nearest integer gives us 13.
 
+Wait, let's double-check the calculations to ensure accuracy.
 
+First, let's compute \(\sqrt{10 - \sqrt{5}}\) more precisely.
 
-## Model Examination [optional]
+\(\sqrt{5} \approx 2.2360679775\), so:
 
-<!-- Relevant interpretability work for the model goes here -->
+\[
+\sqrt{10 - 2.2360679775} = \sqrt{7.7639320225} \approx 2.786
+\]
 
-[More Information Needed]
+Now, compute the expression:
 
-## Environmental Impact
+\[
+-8 \cdot 2.786 + 44 - 4 \cdot 2.236 = -22.288 + 44 - 8.944 = 12.768
+\]
 
-<!-- Total emissions (in grams of CO2eq) and additional considerations, such as electricity usage, go here. Edit the suggested text below accordingly -->
+Rounding 12.768 to the nearest integer gives us 13.
 
-Carbon emissions can be estimated using the [Machine Learning Impact calculator](https://mlco2.github.io/impact#compute) presented in [Lacoste et al. (2019)](https://arxiv.org/abs/1910.09700).
+Thus, the minimum value of the expression is approximately 13.
+</think>
 
-- **Hardware Type:** [More Information Needed]
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-
-## Technical Specifications [optional]
-
-### Model Architecture and Objective
-
-[More Information Needed]
-
-### Compute Infrastructure
-
-[More Information Needed]
-
-#### Hardware
-
-[More Information Needed]
-
-#### Software
-
-[More Information Needed]
-
-## Citation [optional]
-
-<!-- If there is a paper or blog post introducing the model, the APA and Bibtex information for that should go in this section. -->
-
-**BibTeX:**
-
-[More Information Needed]
-
-**APA:**
-
-[More Information Needed]
-
-## Glossary [optional]
-
-<!-- If relevant, include terms and calculations in this section that can help readers understand the model or model card. -->
-
-[More Information Needed]
-
-## More Information [optional]
-
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-
-## Model Card Authors [optional]
-
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-
-## Model Card Contact
-
-[More Information Needed]
+<answer>
+\boxed{13}
+</answer>
+```