Egyptian fractions

Let \(P_1,P_2,P_3, \cdots P_{10}\) be distinct primes, and let \(\displaystyle n=\prod_{i=1}^{10} P_i\). For some \(x,y\in\mathbb{N}\) we have

\[ \frac{1}{n}=\frac{1}{x}+\frac{1}{y}\]

Surprisingly, the number of ordered solutions for this equation is always \(p^k\) for a prime \(p\) and integer \(k\). What is \(p+k\)?

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