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: In a deck of 52 cards, there are 4 suits (hearts, diamonds, clubs, and spades) with 13 cards each. If two cards are drawn randomly without replacement, what is the probability that the first card is a face card (King, Queen, or Jack) and the second card is a number card (Ace through 10)? Express your answer as a common fraction.
Answer: To solve this problem, we need to find the probability of two events:

1. The first card drawn is a face card (King, Queen, or Jack).
2. The second card drawn is a number card (Ace through 10), given that the first card was a face card.

Step 1: Find the probability that the first card is a face card.

There are 12 face cards in a deck of 52 cards (3 face cards per suit $\times$ 4 suits). The probability of drawing a face card first is:
\[ P(\text{face card first}) = \frac{\text{Number of face cards}}{\text{Total number of cards}} = \frac{12}{52} \]

Step 2: Find the probability that the second card is a number card, given that the first card was a face card.

After drawing a face card first, there are 51 cards left in the deck (since we're drawing without replacement), and of those, 36 are number cards (9 number cards per suit $\times$ 4 suits). The probability of drawing a number card second, given that a face card was drawn first, is:
\[ P(\text{number card second | face card first}) = \frac{\text{Number of number cards left}}{\text{Total number of cards left}} = \frac{36}{51} \]

Step 3: Calculate the combined probability of both events happening.

The probability that the first card is a face card and the second card is a number card is the product of the probabilities of each event:
\[ P(\text{face card first and number card second}) = P(\text{face card first}) \times P(\text{number card second | face card first}) = \frac{12}{52} \times \frac{36}{51} \]

Simplify the fraction:
\[ \frac{12}{52} \times \frac{36}{51} = \frac{432}{2652} = \frac{36}{221} \]

So, the answer is:
\[ \boxed{\frac{36}{221}} \]