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 five-digit number is divisible by 6 if and only if the sum of its digits is divisible by 3 and the last digit is even. How many five-digit numbers are there such that the number formed by removing the middle digit is divisible by 6?
Answer: To solve this problem, we can break it down into two main conditions:

1. The sum of the digits (excluding the middle digit) must be divisible by 3.
2. The last digit must be even.

Since we are removing the middle digit, the remaining four-digit number must meet these conditions.

Let's consider the possible sums of the digits that are divisible by 3, for a four-digit number $abcd$ (where $a$, $b$, $c$, and $d$ are the digits):

*   The smallest possible sum is $1+0+0+0 = 1+0+0 = 1+0 = 1$, which is not divisible by 3.
*   The largest possible sum is $9+9+9+9 = 18+9+9 = 27+9 = 36$, which is divisible by 3.

We can see that for every three consecutive integers, one of them will be divisible by 3. So, we can count the number of possible sums as follows:

*   There are 30 possible sums (from 3 to 36, inclusive) that are divisible by 3.

Now, let's count the number of ways to form a four-digit number with an even last digit:

*   There are 5 choices for the last digit ($0$, $2$, $4$, $6$, or $8$).
*   For each of these choices, there are 9 choices for the first digit (since it cannot be 0).
*   For each of these choices, there are 10 choices for the second and third digits.

So, the total number of four-digit numbers with an even last digit is:
\[ 5 \times 9 \times 10 \times 10 = 4500 \]

However, not all of these numbers have a sum of digits that is divisible by 3. To find the number of four-digit numbers that meet both conditions, we can multiply the number of possible sums by the number of ways to form a number with a given sum:

*   For each of the 30 possible sums, there are 150 ways to form a number with that sum (since there are 5 choices for the last digit, 9 choices for the first digit, and 10 choices for the second and third digits, and we can arrange these digits in different ways to get the same sum).

So, the total number of four-digit numbers that meet both conditions is:
\[ 30 \times 150 = 4500 \]

However, this counts some numbers multiple times (since some numbers can have the same sum and last digit). To avoid overcounting, we need to divide by the number of ways to arrange the digits:

*   There are 4! ways to arrange the four digits.

So, the total number of distinct four-digit numbers that meet both conditions is:
\[ 4500 : 4! = 4500 : 24 = 187.5 \]

However, this is not an integer, which means that we have made an error in our counting. Let's re-examine our count:

*   For each of the 30 possible sums, there are actually 50 ways to form a number with that sum (since there are 5 choices for the last digit, and 10 choices for the first, second, and third digits).

So, the total number of four-digit numbers that meet both conditions is:
\[ 30 \times 50 = 1500 \]

To find the number of five-digit numbers that meet the condition, we can multiply the number of four-digit numbers by the number of choices for the middle digit:

*   There are 10 choices for the middle digit.

So, the total number of five-digit numbers that meet the condition is:
\[ 1500 \times 10 = 15000 \]

Therefore, the answer is:
\[ \boxed{15000} \]