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: Let $P(x) = x^3 + ax^2 + bx + c$ be a cubic polynomial with integer coefficients, and $r$, $s$, and $t$ be its roots. If the roots are all integers and $r + s + t = 5$, find the smallest possible value of $r^2 + s^2 + t^2$.
Answer: From Vieta's formulas, we know that the sum of the roots of a polynomial is equal to the negation of the coefficient of the second highest degree term, divided by the leading coefficient. Therefore, for the given cubic polynomial $P(x) = x^3 + ax^2 + bx + c$, the sum of the roots is $-a$.

Given that $r + s + t = 5$, we have $-a = 5$. Since the coefficients are integers, $a$ is an integer, and thus $a = -5$.

Now, we want to minimize $r^2 + s^2 + t^2$. To do this, we need to consider how to distribute the sum of 5 among the three roots $r$, $s$, and $t$ in order to minimize the sum of their squares.

Since the roots are integers, we can think of this as distributing 5 identical items into 3 distinct groups, where each group represents a root. The distribution that minimizes the sum of squares will have the items as evenly distributed as possible.

The most even distribution of 5 items among 3 groups is 1, 2, and 2. This gives us the roots $r = 1$, $s = 2$, and $t = 2$.

Therefore, the smallest possible value of $r^2 + s^2 + t^2$ is:
\[ r^2 + s^2 + t^2 = 1^2 + 2^2 + 2^2 = 1 + 4 + 4 = \boxed{9} \]