Update app.py

app.py

@@ -624,13 +624,21 @@ Please provide:
 5. Finally, if and only if the numerical score is less than 90 out of 100, present a complete revised solution to which you would award a score of 100. 
 The complete revised solution if you provide one must show ALL steps and be fully self-contained.
 
-Important:
+Important Formatting Instructions:
    - Put each solution step on its own line in $$ $$
+   - For inline math expressions, always use $ $, do not use the LaTeX bracket notation, do not use backslash ( ), do not use backslash [ ]
    - DO NOT use begin{{aligned}} or similar environments
    - For currency amounts in dollars, write out the word dollars instead of using $
       * Example: 1000 dollars
    - make sure there are never unmatched dollar signs as that will ruin the LaTeX rendering
 
+Before marking a mathematical expression as incorrect:
+- Check if the expression could be equivalent to what you think it should be through algebraic manipulation
+- Write out the full steps to verify non-equivalence
+- Consider multiple valid ways to write the same mathematical relationship
+- Only mark expressions as wrong if you can prove they are not equivalent to the correct form
+- When grading mathematical solutions:
+
 Format your response with clear headers and bullet points."""
 
         # Call OpenAI API
@@ -639,7 +647,7 @@ Format your response with clear headers and bullet points."""
             messages=[
                 {
                     "role": "assistant",
-                    "content": "I am an expert mathematics professor who grades student solutions using LaTeX formatting. When writing inline math, I always write math using proper LaTeX $ $ delimiters - NEVER the backslash parentheses or backslash brackets approach, NEVER the so-called LaTeX bracket notation. For displayed equations, I use $$ $$."
+                    "content": "I am an expert mathematics professor who grades student solutions using LaTeX formatting. When writing inline math, I always write math using LaTeX $ $ delimiters - NEVER the backslash parentheses or backslash brackets approach, NEVER the so-called LaTeX bracket notation. For displayed equations, I use $$ $$."
                 },
                 {
                     "role": "user",