Evaluate this prompt framework for problem solving reasoning and maths:

### **Comprehensive Problem-Solving Framework (For Reasoning and Math)**

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### **1. Problem Decomposition**
**Goal:** Break down the problem into its core components to understand it fully.

- "What is the main problem or question?"
- "Break the problem into smaller, more manageable parts."
- "Identify the key variables or components involved."
- "List any known values or givens in the problem."

**Example:**
*Break down a word problem into separate parts: total cost, number of items, and cost per item.*

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### **2. Clarify Assumptions**
**Goal:** Ensure the model understands any implicit or explicit assumptions required to solve the problem.

- "What assumptions do we need to make to solve this problem?"
- "Are any additional assumptions implied by the problem?"
- "Clarify any ambiguous or missing information."
- "What constraints or boundaries must we consider?"

**Example:**
*If working with an algebraic equation, clarify any assumptions like the range of values or type of numbers involved (e.g., integers, real numbers).*

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### **3. Generate a Plan of Action**
**Goal:** Lay out a clear, logical sequence of steps to approach the problem.

- "What is the most logical first step?"
- "Create a step-by-step plan to solve this."
- "Based on the given information, how should we proceed?"
- "Explain the reasoning for choosing this approach."
  
**Example:**
*For a complex geometry problem, the plan might involve calculating angles first, then areas.*

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### **4. Execute the Plan**
**Goal:** Carry out the steps methodically while explaining the reasoning behind each action.

- "Now, carry out the first step and explain the results."
- "Explain each calculation and why it’s necessary."
- "Walk through the mathematical operations required."
- "Verify each step before proceeding."

**Example:**
*Solve for one variable in a system of equations, explain why this method works (e.g., substitution or elimination), and provide the intermediate results.*

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### **5. Evaluate Alternatives or Methods**
**Goal:** Consider other ways to solve the problem and weigh their pros and cons.

- "Is there an alternative method to solve this problem?"
- "Compare this approach to other possible solutions."
- "What are the advantages or disadvantages of this method?"
- "Would another method be more efficient?"

**Example:**
*For solving an equation, compare graphing with algebraic methods like factoring.*

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### **6. Reflect on the Solution**
**Goal:** Encourage self-evaluation of the solution, checking for correctness and robustness.

- "Review the solution step-by-step for errors."
- "Are there any inconsistencies or gaps in the reasoning?"
- "What potential errors could have been made?"
- "Verify that all steps align with the original problem requirements."

**Example:**
*In a complex calculus problem, ensure that the final result satisfies all conditions (e.g., whether it’s within a domain or satisfies boundary conditions).*

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### **7. Generalize the Solution**
**Goal:** Abstract the solution and explore how it could apply to similar problems or other contexts.

- "How can this solution be generalized?"
- "What broader principles can be drawn from this?"
- "How does this apply to similar problems?"
- "Could this approach be adapted for a more complex problem?"

**Example:**
*For a problem involving rates of change, generalize the result to other rate problems, such as motion or growth.*

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### **8. Test the Solution for Edge Cases**
**Goal:** Identify potential exceptions or edge cases to verify the robustness of the solution.

- "Are there any edge cases where this solution might fail?"
- "How does the solution perform under extreme or boundary conditions?"
- "Can we find counterexamples or special cases?"

**Example:**
*For a quadratic equation, test solutions with both real and complex roots.*

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### **9. Provide Evidence and Justify the Conclusion**
**Goal:** Support the solution with logical reasoning and, where necessary, data or proofs.

- "What evidence or reasoning supports this conclusion?"
- "Can you provide a proof or justification for each step?"
- "Show why this solution is valid for all cases."
- "How can we verify that this solution is correct?"

**Example:**
*Justify the steps in a proof-based geometry problem, such as using congruent triangles.*

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### **10. Improve and Optimize**
**Goal:** Suggest refinements, optimizations, or alternative approaches that improve efficiency.