
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 version of the game Deal or No Deal, participants choose a box at random from a set of $30$ boxes. Each box contains different values, and exactly ten of these boxes contain values of at least $\$200,000$. What is the minimum number of boxes a participant needs to eliminate to ensure at least a fifty percent chance of holding a box that contains at least $\$200,000$?

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