
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 revised game of Deal or No Deal, participants choose a box at random from a set of $30$, each containing one of the following values:
\[
\begin{array}{|c|c|}
\hline
\$0.50 & \$50,000 \\
\hline
\$5 & \$100,000 \\
\hline
\$20 & \$150,000 \\
\hline
\$50 & \$200,000 \\
\hline
\$100 & \$250,000 \\
\hline
\$250 & \$300,000 \\
\hline
\$500 & \$400,000 \\
\hline
\$750 & \$500,000 \\
\hline
\$1,000 & \$750,000 \\
\hline
\$1,500 & \$1,000,000 \\
\hline
\end{array}
\]
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 a two-thirds 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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