
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  $S_{\pi}=\{i\mid \frac{a_i}{\pi(i)} \text{ is an integer}\}$ . 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  $a_1, \ldots , a_n$ .

*Proposed by James Lin.*

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