Let $\alpha$ be a root of $x^6-x-1$, and call two polynomials $p$ and $q$ with integer coefficients \textit{equivalent} if $p(\alpha)\equiv q(\alpha)\pmod{3}$. It is known that every such polynomial is equivalent to exactly one of $0, 1, x, x^2, \ldots, x^{727}$. Find the largest integer $n < 728$ for which there exists a polynomial $p$ such that $p^3 - p - x^n$ is equivalent to $0$.

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