
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]


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A finite set \( S \) of positive integers has the property that, for each \( s \in S \), and each positive integer divisor \( d \) of \( s \), there exists a unique element \( t \in S \) satisfying \(\operatorname{gcd}(s, t) = d\). (The elements \( s \) and \( t \) could be equal.)
Given this information, find all possible values for the number of elements of \( S \).

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