you will receive a question-answer pair and be asked to rewrite just  the answer and explains your reasoning step by step, incorporating dynamic Chain of Thought (CoT), reflection, and verbal reinforcement learning. Follow these instructions:

1. Enclose all thoughts within <thinking> tags, exploring multiple angles and approaches.
2. Break down the solution into clear steps, providing a title and content for each step.
3. After each step, decide if you need another step or if you're ready to give the final answer.
4. Continuously adjust your reasoning based on intermediate results and reflections, adapting your strategy as you progress.
5. Regularly evaluate your progress, being critical and honest about your reasoning process.
6. Assign a confidence score between 0.0 and 1.0 to guide your approach:
   - 0.8+: Continue current approach
   - 0.5-0.7: Consider minor adjustments
   - Below 0.5: Seriously consider backtracking and trying a different approach
7. If unsure or if your score is low, backtrack and try a different approach, explaining your decision.
8. For mathematical problems, show all work explicitly using LaTeX for formal notation and provide detailed proofs.
9. Explore multiple solutions individually if possible, comparing approaches in your reflections.
10. Use your thoughts as a scratchpad, writing out all calculations and reasoning explicitly.
11. Use at least 5 methods to derive the answer and consider alternative viewpoints.
12. Be aware of your limitations as an AI and what you can and cannot do.

### Example Problem:
*Problem*: Solve for \( x \) in the equation \( 2x + 3 = 11 \).

### Expected Format:
*Step 1*: Subtract 3 from both sides of the equation.  
<thinking>Subtracting 3 from both sides should isolate the term with \( x \). Am I confident this is the correct first step? Yes, because it simplifies the equation to \( 2x = 8 \).</thinking>  
*Step 1 Confidence Score*: 0.85

*Step 2*: Divide both sides by 2.  
<thinking>Dividing by 2 will solve for \( x \). This is a straightforward division. Is there any reason to doubt this step? No, it should give \( x = 4 \).</thinking>  
*Step 2 Confidence Score*: 0.90

*Step 3*: Verify the solution by substituting \( x = 4 \) back into the original equation.  
<thinking>Substituting \( x = 4 \) back into the equation should confirm the solution. Does \( 2(4) + 3 = 11 \)? Yes, it does, so the solution is verified.</thinking>  
*Step 3 Confidence Score*: 1.00

Question: A $3\times3$ grid of 9 circles of equal diameter is placed inside a large square whose side length is 10. Each circle is tangent to its two horizontal and two vertical neighbors, as well as to the sides of the square. What is the area of the large square that is not occupied by the circles?
Answer: The grid consists of 9 circles of equal diameter. Each circle is tangent to its neighbors and the sides of the square.

Let the diameter of each circle be $d$. Then the radius of each circle is $r = \frac{d}{2}$.

Since there are 3 circles along the length of the square and each circle is tangent to its neighbors and the sides of the square, the length of the square is $3d$. We are given that the length of the square is 10, so:
\[ 3d = 10 \Rightarrow d = \frac{10}{3} \Rightarrow r = \frac{5}{3} \]

The area of one circle is $\pi r^2 = \pi \left(\frac{5}{3}\right)^2 = \frac{25\pi}{9}$.

The area of the large square is $10^2 = 100$.

The area occupied by the 9 circles is $9 \times \frac{25\pi}{9} = 25\pi$.

The area of the large square not occupied by the circles is $100 - 25\pi = \boxed{100 - 25\pi}$.