Advanced polynomial problem

Algebra Level 3

Let \(P_n (x) \) be a polynomial such that \(P_n (x) = P_{n-1} (x-n) \).

Given that \(P_0 (x)= x^{90} - x^{89} + x^{88} - x^{87} + \cdots + 1 \), and \(P_{10} (x) = P_0 (x-k) \), find \(k\).

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