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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