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: What is the minimum number of boxes a participant needs to eliminate in order to have a half chance of holding at least $\$200,\!000$ as his or her chosen box, assuming that the participant has already eliminated all the boxes containing less than $\$100,\!000$?
Answer: The participant has eliminated all the boxes containing less than $\$100,\!000$. To have a half chance of holding at least $\$200,\!000$, the participant must eliminate enough boxes so that half of the remaining boxes contain at least $\$200,\!000$.

Let's assume there are $x$ boxes with $\$200,\!000$ or more. The participant needs to eliminate enough boxes so that $x$ is half of the remaining boxes.

Since the participant has already eliminated all boxes with less than $\$100,\!000$, the remaining boxes are those with $\$100,\!000$ or more.

To have a half chance of holding at least $\$200,\!000$, the participant must eliminate all the boxes with exactly $\$100,\!000$, since these boxes do not meet the condition of having at least $\$200,\!000$.

Therefore, the minimum number of boxes the participant needs to eliminate is equal to the number of boxes with exactly $\$100,\!000$.

However, the problem does not provide the exact number of boxes with $\$100,\!000$. To find the minimum number of boxes to eliminate, we need to consider the worst-case scenario.

The worst-case scenario is when there is only one box with $\$200,\!000$ or more, and the rest of the boxes have exactly $\$100,\!000$. In this case, the participant needs to eliminate all the boxes with exactly $\$100,\!000$ to have a half chance of holding the box with at least $\$200,\!000$.

Since there are 26 boxes in total, and 12 of them have less than $\$100,\!000$ (already eliminated), that leaves 14 boxes with $\$100,\!000$ or more.

In the worst-case scenario, 13 out of these 14 boxes have exactly $\$100,\!000$. The participant needs to eliminate these 13 boxes to have a half chance of holding the remaining box with at least $\$200,\!000$.

So, the minimum number of boxes the participant needs to eliminate is:
\[ \boxed{12} \]