
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
Find the smallest positive integer \( n \) that satisfies the following condition: If the set \( A = \{1, 2, 3, \ldots, n\} \) is arbitrarily divided into 63 pairwise disjoint subsets \( A_1, A_2, A_3, \ldots, A_{63} \) (which are non-empty and their union is the set \( A \)), then there always exist two positive integers \( x \) and \( y \) that belong to the same subset \( A_i \) (\( 1 \leq i \leq 63 \)) such that \( x > y \) and \( 31x \leq 32y \).

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