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 $g(x) = \lfloor \log_{10} x \rfloor$ for all positive real numbers $x$. Find the sum of all values of $g(x)$ that satisfy the equation $2g(x) + g(3x) = 10$.
Answer: Given the equation $2g(x) + g(3x) = 10$, we know that $g(x)$ is an integer function because it's defined as the floor of a logarithm. Let's explore the properties of $g(x)$.

$g(x) = \lfloor \log_{10} x \rfloor$ means $g(x)$ is the greatest integer less than or equal to $\log_{10} x$.

Since $2g(x) + g(3x) = 10$, and knowing that $g(x)$ must be an integer, we can consider the possible values of $g(x)$ that could satisfy this equation.

Given that $g(3x)$ is also an integer, the possible combinations of $g(x)$ and $g(3x)$ that sum to 10 are limited. Since $g(x)$ is the floor of a logarithm, it's also important to consider how $g(3x)$ relates to $g(x)$.

For any $x$, $\log_{10} 3x = \log_{10} 3 + \log_{10} x$. Since $\log_{10} 3$ is approximately 0.477, this means that $g(3x)$ will be either equal to $g(x)$ or $g(x) + 1$, depending on the value of $x$.

Let's consider the cases:

1. If $g(3x) = g(x)$, then $2g(x) + g(x) = 10$, which simplifies to $3g(x) = 10$. This has no integer solution for $g(x)$.

2. If $g(3x) = g(x) + 1$, then $2g(x) + g(x) + 1 = 10$, which simplifies to $3g(x) = 9$. This gives $g(x) = 3$.

So, the only possible value for $g(x)$ that satisfies the equation is $g(x) = 3$.

To find the sum of all values of $g(x)$ that satisfy the equation, we only have $g(x) = 3$. Therefore, the sum is simply $3$.

The answer is:
\[ \boxed{3} \]