6. 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 $(2+3)(2+5)(3+5)$.