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 a polynomial $p(x)$ is divided by $(x^2 + 1)$, then the remainder is equal to $ax + b$, where $a$ and $b$ are constants.
Determine $p(x)$ if the remainder of the division of $p(x)$ by $(x^2 + 1)(x - 1)$ is known to be $2x^2 + 5x + 1$.
Answer: Since the remainder is $2x^2 + 5x + 1$, we can write
\[ p(x) = (x^2 + 1)(x - 1)q(x) + 2x^2 + 5x + 1 \]

where $q(x)$ is the quotient.

We also know that if $p(x)$ is divided by $(x^2 + 1)$, the remainder is $ax + b$. This implies
\[ p(x) = (x^2 + 1)r(x) + ax + b \]

where $r(x)$ is the quotient of this division.

Notice that if we substitute $x = 1$ into the first equation, we get
\[ p(1) = (1^2 + 1)(1 - 1)q(1) + 2(1)^2 + 5(1) + 1 \Rightarrow p(1) = 8 \]

Substituting $x = 1$ into the second equation, we get
\[ p(1) = (1^2 + 1)r(1) + a(1) + b \Rightarrow 2r(1) + a + b = 8 \]

Now, to find $a$ and $b$, we need another equation. We can get this by considering the properties of the division.

We know that $p(x) = (x^2 + 1)(x - 1)q(x) + 2x^2 + 5x + 1$. When this is divided by $(x^2 + 1)$, the remainder is $ax + b$.

Since $(x^2 + 1)(x - 1)q(x)$ is divisible by $(x^2 + 1)$ with a remainder of $0$, the remainder of the division of $p(x)$ by $(x^2 + 1)$ is the same as the remainder of the division of $2x^2 + 5x + 1$ by $(x^2 + 1)$.

Let's perform polynomial long division or synthetic division to divide $2x^2 + 5x + 1$ by $x^2 + 1$.

Dividing $2x^2 + 5x + 1$ by $x^2 + 1$ gives us a quotient of $2$ and a remainder of $5x - 1$.

Therefore, $ax + b = 5x - 1$, so $a = 5$ and $b = -1$.

To find $p(x)$, we can use the equation
\[ p(x) = (x^2 + 1)(x - 1)q(x) + 2x^2 + 5x + 1 \]

Since we don't know $q(x)$, we can't find $p(x)$ exactly, but we can express it in terms of $q(x)$.

However, we can also express $p(x)$ using the given remainder $ax + b$ as
\[ p(x) = (x^2 + 1)r(x) + 5x - 1 \]

Since we don't have enough information to determine $q(x)$ or $r(x)$, we can't find a specific $p(x)$. However, based on the question, it seems we are looking for a general form or a specific case that satisfies the given conditions.

In the general case, $p(x)$ can be any polynomial that, when divided by $(x^2 + 1)(x - 1)$, gives a remainder of $2x^2 + 5x + 1$. One such polynomial is
\[ p(x) = (x^2 + 1)(x - 1)q(x) + 2x^2 + 5x + 1 \]

This is not a specific polynomial but a form that $p(x)$ can take.

Given the remainder $2x^2 + 5x + 1$, one possible polynomial is
\[ p(x) = 2x^2 + 5x + 1 \]

This polynomial, when divided by $(x^2 + 1)(x - 1)$, gives a remainder of $2x^2 + 5x + 1$, and when divided by $(x^2 + 1)$, gives a remainder of $5x - 1$, satisfying the conditions given.

Therefore, one possible answer for $p(x)$ is:
\[ p(x) = \boxed{2x^2 + 5x + 1} \]