
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  $S$  be a nonempty set of primes satisfying the property that for each proper subset  $P$  of  $S$ , all the prime factors of the number  $\left(\prod_{p\in P}p\right)-1$  are also in  $S$ . Determine all possible such sets  $S$ .

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