Update app.py

app.py

@@ -705,24 +705,34 @@ def perform_final_verification(revised_solution):
 Revised Solution to Verify:
 {revised_solution}
 
-Please:
-1. Verify the mathematical correctness of the solution
-2. Check for any subtle errors or missing cases
-3. Verify that all assumptions are clearly stated
-4. Ensure the solution is complete and rigorous
-5. Check that any series convergence, existence conditions, or boundary cases are properly addressed
-
-If you find any issues:
-1. Clearly explain what is incorrect or missing
-2. Provide a complete corrected solution
-3. Maintain the same LaTeX formatting as the original
-4. Include any missing assumptions or conditions
-5. If relevant, provide corrected SymPy code
-
-If the solution is correct:
-Briefly explain why the solution is mathematically sound and complete.
-
-Please ensure any corrected solution maintains proper LaTeX formatting with $ for inline math and $$ on separate lines for displayed equations."""
+Please follow these steps exactly:
+
+1. First, analyze the solution for:
+   - Mathematical correctness
+   - Missing cases or assumptions
+   - Completeness and rigor
+   - Necessary conditions and edge cases
+   - Any subtle errors or oversights
+
+2. After your analysis, write exactly this phrase:
+   "Here is the complete verified solution:"
+
+3. Then write out the ENTIRE solution from beginning to end, including:
+   - All correct parts from the original solution
+   - All needed corrections
+   - All additional cases and verifications
+   - Any missing steps or assumptions
+   - Any necessary additional proofs or derivations
+
+Your complete solution must:
+- Be completely self-contained
+- Not refer to the original solution
+- Show every step of the calculation
+- Include all necessary verifications
+- Maintain proper LaTeX formatting with $ for inline math and $ on separate lines
+- Include any relevant SymPy code for verification
+
+Remember to write out the complete solution even if you only need to add a few steps - the goal is to have a single, complete, verified solution that includes everything necessary for full mathematical rigor."""
 
     try:
         # Make API call for final verification