Let $P(x) = x^3 - 6x^2 - 5x + 4$. Suppose that $y$ and $z$ are real numbers such that \[ zP(y) = P(y - n) + P(y + n) \] for all real numbers $n$. Evaluate $P(y)$.

Let's think step by step and output the final answer within \boxed{}.
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