
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
Let \( n \) be a fixed positive integer, and choose \( n \) positive integers \( a_{1}, \ldots, a_{n} \). Given a permutation \( \pi \) on the first \( n \) positive integers, let \( S_{\pi}=\left\{i \left\lvert\, \frac{a_{i}}{\pi(i)}\right.\right. \text{ is an integer} \} \). Let \( N \) denote the number of distinct sets \( S_{\pi} \) as \( \pi \) ranges over all such permutations. Determine, in terms of \( n \), the maximum value of \( N \) over all possible values of \( a_{1}, \ldots, a_{n} \).

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