
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
Determine all nonempty finite sets \( S=\{a_{1}, \ldots, a_{n}\} \) of \( n \) distinct positive integers such that \( a_{1} \cdots a_{n} \) divides \((x+a_{1}) \cdots (x+a_{n})\) for every positive integer \( x \).

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