
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  $n \ge 3$  be a positive integer. Suppose that  $\Gamma$  is a unit circle passing through a point  $A$ . A regular  $3$ -gon, regular  $4$ -gon, \dots, regular  $n$ -gon are all inscribed inside  $\Gamma$  such that  $A$  is a common vertex of all these regular polygons. Let  $Q$  be a point on  $\Gamma$  such that  $Q$  is a vertex of the regular  $n$ -gon, but  $Q$  is not a vertex of any of the other regular polygons. Let  $\mathcal{S}_n$  be the set of all such points  $Q$ . Find the number of integers  $3 \le n \le 100$  such that
\[
        \prod_{Q \in \mathcal{S}_n} |AQ| \le 2.
 \]

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