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: There are exactly three integers $x$ satisfying the equation

$$x^2+cx+d=0.$$

Find the number of possible values of $c$ if $d$ is a prime number.
Answer: For a quadratic equation of the form $x^2 + cx + d = 0$ to have exactly three integer solutions, the discriminant $\Delta = c^2 - 4d$ must be a perfect square.

Additionally, the prime number $d$ must be a factor of the product of the roots, which are integers.

Since the sum of the roots is $-c$ and the product of the roots is $d$, and given that there are exactly three integer solutions, we can infer that one of the solutions must be repeated.

Let the roots be $r$, $r$, and $s$. We have the following relationships:
   \begin{align*}
   r + r + s &= -c \\
   rr + rs + rs &= d
   \end{align*}

Simplifying the relationships, we get:
   \begin{align*}
   2r + s &= -c \\
   r^2 + 2rs &= d
   \end{align*}

Since $d$ is prime, $r^2 + 2rs = d$ implies that either $r^2 = 1$ and $2rs = d - 1$, or $r^2 = d$ and $s = 0$.

Considering $r^2 = 1$, we have $r = \pm 1$. If $r = 1$, then $2s = d - 1$, and since $d$ is prime, $d - 1$ must be even, which is only possible if $d = 2$.

If $r = -1$, then $-2s = d - 1$, leading to the same conclusion that $d = 2$.

Considering $r^2 = d$, since $d$ is prime, $r^2$ cannot be a perfect square unless $d = 2$.

Thus, the only possible value for $d$ is 2, and the corresponding values for $r$ are 1 and -1.

For each value of $r$, we can find the corresponding value of $s$ and subsequently $c$. Since $2r + s = -c$, and we have two possible values for $r$, there are two possible values for $c$.

Therefore, the number of possible values of $c$ is $\boxed{2}$.