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: The operation $*$ is defined as $a * b = 2a - 2b$. If $x * (x + 2) = -4$, what is the value of $x$?
Answer: We have $x * (x + 2) = -4$, and we know that $*$ is defined as $a * b = 2a - 2b$. Therefore, we can write the equation as:
\[ 2x - 2(x + 2) = -4 \]

Expanding the equation, we get:
\[ 2x - 2x - 4 = -4 \Rightarrow -4 \]

This equation is always true, which means that the value of $x$ can be any number. However, looking at the original equation, we see that it can be rewritten as:
\[ x * (x + 2) = 2x - 2(x + 2) = 2x - 2x - 4 = -4 \]

This equation does not depend on $x$, so any value of $x$ will satisfy the equation.

However, looking at the definition of the operation $*$ again, we see that the result of $a * b$ is $2a - 2b$. This means that the result of $x * (x + 2)$ is $2x - 2(x + 2) = -4$.

This equation can be simplified to $-4$, which is always true. However, this does not give us a specific value for $x$.

Since the equation $x * (x + 2) = -4$ is always true, and the result of the operation $x * (x + 2)$ is always $-4$, regardless of the value of $x$, we can say that the value of $x$ is not fixed.

However, if we really need to provide a value for $x$, we can try to find a value that makes the equation true.

One possible approach is to try to find a value of $x$ that makes the equation $2x - 2(x + 2) = -4$ true.

To do this, we can simplify the equation:
\[ 2x - 2x - 4 = -4 \Rightarrow -4 \]

As we can see, this equation is always true, regardless of the value of $x$.

However, if we want to provide a specific value for $x$, we can say that $x$ can be any number, for example, $x = 0$.

Therefore, the value of $x$ is:
\[ \boxed{0} \]