elementary number theory - What is the largest three-digit integer that when cubed, the result ends in itself

Let $N =\overline{abc}$ be a three-digit integer with distinct digits $a$, $b$, and $c$. What is the largest possible integer $N$ such that, when $N$ is cubed, the resulting integer ends with the same three digits as $N$?

Here is what I did:
I know that $N^3\equiv N \pmod{1000}.$ That means that $N^3-N\equiv 0 \pmod{1000}$ or $N(N-1)(N+1)\equiv0 \pmod{1000}.$ However, I don't know how to quickly find numbers that fit the properties without brute force. What do I do?