Do You Like "Odd" Primes?

Let $$P$$ denote the set of odd primes under 1000. $$p$$ is an element of $$P$$ that satisfies the following:

For any $$S=\{p_1,p_2,\ldots,p_k\}\subseteq P$$ where $$k\ge 2$$ and $$p_i\ne p$$ for $$i=1,2,\ldots,k$$, there exists $$q\in P\backslash S$$ such that

$q+1|(p_1+1)(p_2+1)\cdots(p_k+1).$

Find the sum of all possible $$p$$.

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