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: Find the number of positive integers less than or equal to 1000 that are divisible by 7 and have exactly 4 positive divisors.
Answer: To solve this problem, we need to find the number of positive integers less than or equal to 1000 that are divisible by 7 and have exactly 4 positive divisors.

A positive integer $n$ has exactly 4 positive divisors if and only if $n$ is the product of two distinct prime numbers or $n$ is the cube of a prime number.

We need to count the number of positive integers less than or equal to 1000 that are divisible by 7 and satisfy one of the above conditions.

Let's first count the number of positive integers less than or equal to 1000 that are divisible by 7 and are the product of two distinct prime numbers.

The smallest prime number greater than 7 is 11. Therefore, the smallest positive integer that is divisible by 7 and is the product of two distinct prime numbers is $7 \cdot 11 = 77$.

The largest prime number less than or equal to $\frac{1000}{7}$ is 139. Therefore, the largest positive integer that is divisible by 7 and is the product of two distinct prime numbers is $7 \cdot 139 = 973$.

Thus, the number of positive integers less than or equal to 1000 that are divisible by 7 and are the product of two distinct prime numbers is equal to the number of prime numbers between 11 and 139, inclusive.

There are 32 prime numbers between 11 and 139, inclusive.

Next, let's count the number of positive integers less than or equal to 1000 that are divisible by 7 and are the cube of a prime number.

The smallest prime number is 2, and $2^3 = 8$. However, 8 is not divisible by 7.

The next prime number is 3, and $3^3 = 27$. However, 27 is not divisible by 7.

The next prime number is 5, and $5^3 = 125$. However, 125 is not divisible by 7.

The next prime number is 7, and $7^3 = 343$. This is the only cube of a prime number that is divisible by 7 and less than or equal to 1000.

Therefore, there is only 1 positive integer less than or equal to 1000 that is divisible by 7 and is the cube of a prime number.

In total, there are $32 + 1 = 33$ positive integers less than or equal to 1000 that are divisible by 7 and have exactly 4 positive divisors.

Thus, the answer is:
\[ \boxed{33} \]