Another mean question

Number Theory Level pending

Let \(S\) be a set of (distinct!) prime numbers, and let \(\langle S\rangle_G\) be the geometric mean of \(S\):

\[\langle S\rangle_G = \sqrt[\#S]{\prod S}.\]

Suppose there exist a prime number \(p\) and a positive integer \(n\) such that \(\langle S \rangle_G^n = p\). What is the greatest possible number of elements in \(S\)?

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