How Far Can You Go?

Find the sum of all positive integers \(n\) such that there exists \(p\in \mathbb{N}\) that satisfies the divisibility

\[p^{n-1}+(p+2)^{n-1}\mid p^n+(p+2)^n.\]

This problem is a generalization by me of a problem that appeared in St. Petersburg City Mathematical Olympiad 1996.

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