
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In a modified game of Deal or No Deal, participants choose a box at random from a set of $30$ boxes, each containing one of the following values: \begin{tabular}{|c|c|}\hline\$.01&\$2,000\\\hline\$1&\$10,000\\\hline\$10&\$50,000\\\hline\$100&\$100,000\\\hline\$500&\$200,000\\\hline\$1,000&\$300,000\\\hline\$5,000&\$400,000\\\hline\$7,500&\$500,000\\\hline\$10,000&\$750,000\\\hline\$15,000&\$1,000,000\\\hline\end{tabular} After choosing a box, participants eliminate other boxes by opening them. What is the minimum number of boxes a participant needs to eliminate in order to have at least a half chance of holding a box with $\$200,\!000$ or more?

Present the answer in LaTex format: \boxed{Your answer}
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