
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 integers \( n \geq 3 \) such that

\[ n! \mid \prod_{\substack{p<q \leq n \\ p, q \text { prime }}}(p+q) \]

Remark: The expression on the right-hand side denotes the product over all sums of two distinct primes less than or equal to \( n \). For \( n=6 \), this is equal to \( (2+3)(2+5)(3+5) \).

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