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: If $x^2+bx+9$ has two non-real roots, then the discriminant of the quadratic equation is negative, i.e., $b^2-4ac<0$, where $a=1$ and $c=9$. Therefore, we have

$$b^2-4(1)(9)<0\Rightarrow b^2<36\Rightarrow |b|<6\Rightarrow -6<b<6.$$

Thus, all real possible values of $b$ are in the interval $(-6,6)$.

Problem:
Determine the value of $k$ such that the equation

$$x^2-3x+k=0$$

has two real roots with the same absolute value but opposite signs.

Solution:
If the equation has two real roots with the same absolute value but opposite signs, then the product of the roots must be negative, i.e., $\frac{c}{a}<0$, where $a=1$ and $c=k$. Therefore, we have $k<0$. Furthermore, the sum of the roots must be zero, i.e., $-\frac{b}{a}=0$, where $a=1$ and $b=-3$. Since $b\neq 0$, this is not possible. Thus, there is no value of $k$ such that the equation has two real roots with the same absolute value but opposite signs.
Answer: The roots of the quadratic equation $x^2 - 3x + k = 0$ are $x_1$ and $x_2$. We know that if the roots have the same absolute value but opposite signs, then the product of the roots is negative and the sum of the roots is zero.

1. The product of the roots is given by:
\[ x_1 x_2 = k \]

Since the product is negative, we have:
\[ k < 0 \]

2. The sum of the roots is given by:
\[ x_1 + x_2 = 3 \]

Since the sum must be zero, we have a contradiction because $x_1 + x_2 = 3 \neq 0$.

Therefore, there is no value of $k$ that satisfies the conditions.

In other words, the answer is:
\[ \boxed{\text{No solution}} \]