How to instruct the model to follow this framewok, doing the thinking and evaluation process internally, and in the output to present only the final solution? REWRITE THE FRAMEWORK:
Here’s a structured **prompt framework** tailored for solving problems in reasoning, logic, or mathematics. It 

incorporates critical thinking elements, step-by-step decomposition, comparative analysis, and self-reflection, 

using the keyword techniques discussed.

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

---

### **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.*

---

### **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).*

---

### **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.*

---

### **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.*

---

### **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.*

---

### **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).*

---

### **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.*

---

### **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.*

---

### **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.*

---

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

- "Is this solution the most efficient?"
- "Can the solution be improved or optimized?"
- "What would make this approach faster or simpler?"
- "Can this solution be refined for more difficult problems?"

**Example:**
*In an optimization problem, refine the solution to minimize costs or maximize output with different constraints.*

---

### **Example of Applying the Framework:**

**Prompt:**
"I have a word problem involving two trains traveling towards each other from different cities. One train is 

moving at 80 km/h and the other at 60 km/h. They start 280 km apart. How long will it take for them to meet? 

Please use a step-by-step approach to break down the problem, identify key assumptions, calculate the time, 

and verify the solution."

**Response Framework Application:**
1. **Problem Decomposition**: Break down the components—train speeds, distance, time to meet.
2. **Clarify Assumptions**: Assume constant speed for both trains and no stops.
3. **Plan of Action**: Add the speeds together (80 km/h + 60 km/h) and use distance = speed × time.
4. **Execute the Plan**: 280 km ÷ 140 km/h = 2 hours.
5. **Evaluate Alternatives**: Could verify by graphing the distance over time for both trains.
6. **Reflect on the Solution**: Check if 2 hours makes sense given the total distance and relative speeds.
7. **Generalize**: Apply the same method to other two-object motion problems.
8. **Edge Cases**: Consider cases where one train moves faster or starts closer.
9. **Justify**: Time = distance ÷ speed is mathematically sound.
10. **Improve**: For more complex scenarios, include acceleration or variable speeds.