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

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