TW-TST 1999 Problem 4

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 $(p_{1}+1)(p_{2}+1)...(p_{k}+1)$ .