
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  $a_1 < a_2 < \cdots < a_k$  denote the sequence of all positive integers between  $1$  and  $91$  which are relatively prime to  $91$ , and set  $\omega = e^{2\pi i/91}$ . Define \[S = \prod_{1\leq q < p\leq k}\left(\omega^{a_p} - \omega^{a_q}\right).\] Given that  $S$  is a positive integer, compute the number of positive divisors of  $S$ .

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