
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

Given the set \( A = \{1, 2, 3, \ldots, 1002\} \), Petya and Vasya play a game. Petya names a number \( n \), and Vasya then selects a subset of \( A \) consisting of \( n \) elements. Vasya wins if there are no two coprime numbers in the subset he selects; otherwise, Petya wins. What is the minimum \( n \) that Petya must name to guarantee a win?

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