
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 the game Deal or No Deal, participants choose a box at random from a set of $30,$ each containing one of the following values: \begin{tabular}{|c|c|}\hline\$.01&\$1,000\\\hline\$1&\$5,000\\\hline\$5&\$10,000\\\hline\$10&\$50,000\\\hline\$25&\$75,000\\\hline\$50&\$100,000\\\hline\$75&\$200,000\\\hline\$100&\$300,000\\\hline\$200&\$400,000\\\hline\$300&\$500,000\\\hline\$400&\$600,000\\\hline\$500&\$700,000\\\hline\$750&\$1,000,000\\\hline\end{tabular} After choosing a box, participants eliminate other boxes by opening them and then removing that box and its amount from the game. 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?

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