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: The lines $2y + x + 3 = 0$ and $3y + ax + 2 = 0$ are perpendicular if and only if their slopes are negative reciprocals of each other. The slope of the first line is $-\frac{1}{2}$, and the slope of the second line is $-\frac{a}{3}$. Therefore, we must have $-\frac{a}{3} = \frac{2}{1}$. Solving for $a$, we get $a = -6$.

Another problem inspired by this one is:

Find the equation of the line that passes through the point $(1, 2)$ and is perpendicular to the line $y = 2x - 3$.

This problem requires a different approach to solve, as it involves finding the equation of a line given a point and a slope, rather than just finding the slope of a line. To solve this problem, we can use the point-slope form of a line, which is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope. In this case, we know that the line passes through the point $(1, 2)$ and has a slope that is the negative reciprocal of the slope of the line $y = 2x - 3$. The slope of the line $y = 2x - 3$ is $2$, so the slope of the perpendicular line is $-\frac{1}{2}$. Plugging these values into the point-slope form, we get $y - 2 = -\frac{1}{2}(x - 1)$. Simplifying this equation, we get $y = -\frac{1}{2}x + \frac{5}{2}$, which is the equation of the line that passes through the point $(1, 2)$ and is perpendicular to the line $y = 2x - 3$.
Answer: We are given the equation of a line $y = 2x - 3$. We are asked to find the equation of a line that passes through the point $(1, 2)$ and is perpendicular to the line $y = 2x - 3$.

First, let's find the slope of the given line:
\[ y = 2x - 3 \Rightarrow \text{slope} = 2 \]

The slope of the perpendicular line will be the negative reciprocal of the slope of the given line:
\[ \text{slope of perpendicular line} = -\frac{1}{2} \]

Now, we can use the point-slope form of a line to find the equation of the perpendicular line:
\[ y - y_1 = m(x - x_1) \]

where $(x_1, y_1)$ is the point $(1, 2)$ and $m$ is the slope $-\frac{1}{2}$.

Substituting these values, we get:
\[ y - 2 = -\frac{1}{2}(x - 1) \]

Simplifying the equation, we get:
\[ y - 2 = -\frac{1}{2}x + \frac{1}{2} \Rightarrow y = -\frac{1}{2}x + \frac{1}{2} + 2 \Rightarrow y = -\frac{1}{2}x + \frac{5}{2} \]

So, the equation of the line that passes through the point $(1, 2)$ and is perpendicular to the line $y = 2x - 3$ is:
\[ \boxed{y = -\frac{1}{2}x + \frac{5}{2}} \]