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: Given that the two roots of the cubic equation $ax^3 + (a + 2b) x^2 + (b - 3a) x + (8 - a) = 0$ are $-2$ and $3$, find the values of $a$ and $b$.

Let's start by using the fact that if $x = -2$ and $x = 3$ are roots, then $(x + 2)$ and $(x - 3)$ are factors of the cubic polynomial. We can write the polynomial as:

$$ax^3 + (a + 2b) x^2 + (b - 3a) x + (8 - a) = a(x + 2)(x - 3)(x - r)$$

where $r$ is the third root. Expanding the right-hand side and comparing coefficients, we get:

$$a(x^2 - x - 6)(x - r) = ax^3 + (a + 2b) x^2 + (b - 3a) x + (8 - a)$$

Comparing the coefficients of $x^3$, we get $a = a$. Comparing the coefficients of $x^2$, we get $-a - 6a = a + 2b$, which simplifies to $-8a = 2b$ or $b = -4a$. Comparing the constant terms, we get $-6ar = 8 - a$.

Substituting $b = -4a$ into the original equation, we get:

$$ax^3 + (a - 8a) x^2 + (-4a - 3a) x + (8 - a) = 0$$

Simplifying, we get:

$$ax^3 - 7ax^2 - 7ax + (8 - a) = 0$$

Now, we can use the fact that $x = -2$ and $x = 3$ are roots to find the value of $a$. Plugging in $x = -2$, we get:

$$a(-2)^3 - 7a(-2)^2 - 7a(-2) + (8 - a) = 0$$

Simplifying, we get $-8a - 28a + 14a + 8 - a = 0$, which simplifies to $-23a + 8 = 0$. Solving for $a$, we get $a = 8/23$.

Plugging in $x = 3$, we get:

$$a(3)^3 - 7a(3)^2 - 7a(3) + (8 - a) = 0$$

Simplifying, we get $27a - 63a - 21a + 8 - a = 0$, which simplifies to $-58a + 8 = 0$. Solving for $a$, we get $a = 8/58 = 4/29$.

Since we got different values for $a$, this means that the original equation does not have a solution for $a$ and $b$. Therefore, the third root cannot be found.

So, the new problem is:

Given that the two roots of the cubic equation $ax^3 + bx^2 + cx + d = 0$ are $r_1$ and $r_2$, and the coefficients $a, b, c, d$ are related by a certain condition, find the values of $a, b, c, d$.

Note that this problem is more general than the original one and requires a different approach to solve.
Answer: There is no solution for $a$ and $b$ because the system of equations is inconsistent.

Now it's your turn! Here is a new question for you to solve:

A polynomial $P(x)$ of degree 4 is such that $P(1) = P(2) = P(3) = 0$, and $P(0) = 1$. What is the value of $P(4)$?

(Remember to put your answer inside \boxed{}.)