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: Let $a_n$ be a sequence such that $a_1 = 1$, $a_2 = 1$, and $a_n = (n-1) \cdot a_{n-1} + a_{n-2}$ for $n \geq 3$.
Find the remainder when $a_{2006}$ is divided by $100$.
Answer: To find the remainder of $a_{2006}$ when divided by $100$, we can start by calculating the first few terms of the sequence modulo $100$. Given the recurrence relation $a_n = (n-1) \cdot a_{n-1} + a_{n-2}$ for $n \geq 3$, with initial conditions $a_1 = 1$ and $a_2 = 1$, we can proceed to calculate the terms modulo $100$.

\[ a_3 = (3-1) \cdot a_2 + a_1 = 2 \cdot 1 + 1 = 3 \]

\[ a_4 = (4-1) \cdot a_3 + a_2 = 3 \cdot 3 + 1 = 10 \]

\[ a_5 = (5-1) \cdot a_4 + a_3 = 4 \cdot 10 + 3 = 43 \]

\[ a_6 = (6-1) \cdot a_5 + a_4 = 5 \cdot 43 + 10 = 225 \equiv 25 \mod 100 \]

\[ a_7 = (7-1) \cdot a_6 + a_5 = 6 \cdot 25 + 43 = 193 \equiv 93 \mod 100 \]

\[ a_8 = (8-1) \cdot a_7 + a_6 = 7 \cdot 93 + 25 = 676 \equiv 76 \mod 100 \]

\[ a_9 = (9-1) \cdot a_8 + a_7 = 8 \cdot 76 + 93 = 701 \equiv 1 \mod 100 \]

\[ a_{10} = (10-1) \cdot a_9 + a_8 = 9 \cdot 1 + 76 = 85 \]

\[ a_{11} = (11-1) \cdot a_{10} + a_9 = 10 \cdot 85 + 1 = 851 \equiv 51 \mod 100 \]

\[ a_{12} = (12-1) \cdot a_{11} + a_{10} = 11 \cdot 51 + 85 = 746 \equiv 46 \mod 100 \]

From the pattern observed, the sequence repeats every $10$ terms starting from $a_3$ (with the terms $a_3$ to $a_{12}$ forming the first cycle of length $10$), and then continues in a similar manner. Therefore, to find $a_{2006} \mod 100$, we need to find the position of $a_{2006}$ within the repeating cycle.

Since the sequence repeats every $10$ terms and $2006$ is $6$ more than a multiple of $10$, $a_{2006}$ will be equivalent to $a_6$ modulo $100$.

Hence, the remainder when $a_{2006}$ is divided by $100$ is the same as the remainder when $a_6$ is divided by $100$, which is:
\[ \boxed{25} \]