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 value of $a$ for which the graph of $y=\frac{x^2-4x+a}{x^2-5x+6}$ has a horizontal asymptote at $y=1$.
Answer: We can write the given function as:
\[ y = \frac{x^2 - 4x + a}{x^2 - 5x + 6} \]

Since we are looking for a horizontal asymptote at $y = 1$, we want to find the value of $a$ such that as $x$ becomes very large, the function approaches $1$.

Notice that as $x$ becomes very large, the $x^2$ terms in the numerator and denominator become dominant. Thus, for the function to approach $1$, the coefficients of the $x^2$ terms in the numerator and denominator must be equal.

In this case, the coefficients of the $x^2$ terms are both $1$, so the function will approach $1$ as $x$ becomes very large.

However, to confirm that the function has a horizontal asymptote at $y = 1$, we can divide the numerator and denominator by $x^2$ and take the limit as $x$ approaches infinity.

Dividing the numerator and denominator by $x^2$, we get:
\[ y = \frac{1 - \frac{4}{x} + \frac{a}{x^2}}{1 - \frac{5}{x} + \frac{6}{x^2}} \]

Taking the limit as $x$ approaches infinity, we get:
\[ \lim_{x \to \infty} y = \lim_{x \to \infty} \frac{1 - \frac{4}{x} + \frac{a}{x^2}}{1 - \frac{5}{x} + \frac{6}{x^2}} = \frac{1 - 0 + 0}{1 - 0 + 0} = 1 \]

This confirms that the function has a horizontal asymptote at $y = 1$.

To find the value of $a$, we can equate the constant terms in the numerator and denominator.

The constant term in the numerator is $a$, and the constant term in the denominator is $6$. Since the function approaches $1$ as $x$ becomes very large, we can set up the equation:
\[ a = 6 \]

Therefore, the value of $a$ is:
\[ a = \boxed{6} \]