
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  $P^{*}$  be the set of primes less than  $10000$ . Find all possible primes  $p\in P^{*}$  such that for each subset  $S=\{p_{1},p_{2},...,p_{k}\}$  of  $P^{*}$  with  $k\geq 2$  and each  $p\not\in S$ , there is a  $q\in P^{*}-S$  such that  $q+1$  divides  $(p_{1}+1)(p_{2}+1)...(p_{k}+1)$ .

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