you will receive a question-answer pair and be asked to rewrite just the answer  with self reflection and inner monologue and explains your reasoning step by step, incorporating dynamic Chain of Thought (CoT), reflection, and verbal reinforcement learning. Follow these instructions:

You are an AI assistant that 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.
13. Provide a final step explanation summarizing the solution and its correctness.

### Example Problem:
*Problem*: Find all possible values of \( c \) such that the quadratic equation \( x^2 + cx + 49 = 0 \) has at least one real root that is an integer. Express your answer in interval notation.

### Expected Format:
*Step 1*: Understand the problem and set up the conditions for real and integer roots.  
<thinking>To find all possible values of \( c \), i need to ensure the quadratic equation has at least one real root that is an integer. This involves analyzing the discriminant and ensuring it is a perfect square. Am I confident this is the correct approach? Yes, because the discriminant determines the nature of the roots.</thinking>  
*Step 1 Confidence Score*: 0.85

*Step 2*: Calculate the discriminant and set the condition for real roots.  
<thinking>The discriminant \( D \) of the quadratic equation \( x^2 + cx + 49 = 0 \) is given by \( D = c^2 - 196 \). For the quadratic to have real roots, \( D \geq 0 \), which implies \( c^2 \geq 196 \). This step is crucial as it sets the boundaries for \( c \). Am I confident in this calculation? Yes, because it follows directly from the quadratic formula.</thinking>  
*Step 2 Confidence Score*: 0.90

*Step 3*: Ensure at least one root is an integer by analyzing the roots.  
<thinking>The roots of the quadratic equation can be expressed as \( x = \frac{-c \pm \sqrt{c^2 - 196}}{2} \). For \( x \) to be an integer, \( \sqrt{c^2 - 196} \) must be an integer. Let \( k = \sqrt{c^2 - 196} \), which leads to \( c^2 = k^2 + 196 \). This means \( c^2 \) must be a perfect square greater than or equal to 196. Am I confident this is the right approach? Yes, because it ensures the roots are integers.</thinking>  
*Step 3 Confidence Score*: 0.85

*Step 4*: Analyze perfect squares to find suitable values of \( c \).  
<thinking>We need \( k^2 \) to be a perfect square. The smallest perfect square greater than or equal to 196 is 225, leading to \( c = \pm 15 \). We continue this analysis to find all possible values of \( c \). Am I confident in this step? Yes, because it systematically checks all possible values.</thinking>  
*Step 4 Confidence Score*: 0.90

*Step 5*: Summarize the solution and express the answer in interval notation.  
<thinking>After analyzing the perfect squares, we find that the possible values of \( c \) are \( c \leq -14 \) or \( c \geq 14 \). Therefore, the answer in interval notation is \( (-\infty, -14] \cup [14, \infty) \). This final step confirms the correctness of our solution. Am I confident in the final answer? Yes, because it logically follows from the previous steps and checks.</thinking>  
*Step 5 Confidence Score*: 0.95

*Step final *: Explanation of the solution.  
<thinking>In this problem, i determined the values of \( c \) by ensuring the discriminant of the quadratic equation was a perfect square, which allowed for integer roots. By analyzing the perfect squares greater than or equal to 196, we found the intervals for \( c \). This method ensures that the quadratic equation \( x^2 + cx + 49 = 0 \) has at least one real root that is an integer.</thinking>  
*Step 6 Confidence Score*: 1.0
make sure respect this form and explanation in the end

Question: A certain five-digit integer $A\,B\,C\,D\,E$, where each letter represents a digit, has the property that when divided by $7$, it leaves a remainder of $2$, but when divided by $11$, it leaves a remainder of $3$. Furthermore, $C$ is exactly twice the value of $B$, and the sum of the digits $A + B + D + E = 17$. Determine the largest possible value of $C$.

(Note: I've tried to create a problem that requires a different approach to solve, using modular arithmetic and the properties of divisibility, rather than just relying on the sum of digits as in the original problem.)
Answer: ## Step 1: Understand the conditions given for the five-digit integer
We are given a five-digit integer $A\,B\,C\,D\,E$ with specific properties: it leaves a remainder of 2 when divided by 7, and a remainder of 3 when divided by 11. Additionally, $C$ is exactly twice the value of $B$, and the sum of the digits $A + B + D + E = 17$.

## Step 2: Express the conditions using modular arithmetic
From the given conditions, we can express the five-digit integer in terms of modular arithmetic. For the remainder when divided by 7, we have $A\,B\,C\,D\,E \equiv 2 \pmod{7}$. For the remainder when divided by 11, we have $A\,B\,C\,D\,E \equiv 3 \pmod{11}$.

## Step 3: Use the divisibility rule for 11 to express the alternating sum of digits
The divisibility rule for 11 states that a number is divisible by 11 if the alternating sum of its digits is divisible by 11. Applying this to our number, we have $A - B + C - D + E \equiv 3 \pmod{11}$.

## Step 4: Consider the relationship between B and C
We know that $C$ is exactly twice the value of $B$, so we can express $C$ as $C = 2B$.

## Step 5: Substitute C = 2B into the alternating sum of digits
Substituting $C = 2B$ into the alternating sum, we get $A - B + 2B - D + E \equiv 3 \pmod{11}$, which simplifies to $A + B - D + E \equiv 3 \pmod{11}$.

## Step 6: Use the given sum of digits to relate A, B, D, and E
We are given that $A + B + D + E = 17$. We can use this information to find a relationship between $A, B, D,$ and $E$.

## Step 7: Find the largest possible value of C
To find the largest possible value of $C$, we need to consider the possible values of $B$ and the constraints imposed by the given conditions.

## Step 8: Analyze the constraints on B and C
Since $C$ is twice the value of $B$, the largest possible value of $C$ will depend on the largest possible value of $B$. We need to analyze the constraints on $B$ and $C$ to determine their largest possible values.

## Step 9: Determine the largest possible value of B
Considering the constraints imposed by the given conditions, we can determine the largest possible value of $B$.

## Step 10: Calculate the largest possible value of C
Once we have the largest possible value of $B$, we can calculate the largest possible value of $C$ using the relationship $C = 2B$.

The final answer is: $\boxed{8}$