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Algebra Level 4

Suppose $f(x)$ is a monic integer polynomial of degree three. Find the largest possible number of distinct integers $n$ such that $f(n^2)=f(n)$ .

Details and assumptions

A polynomial is monic if its leading coefficient is 1. For example, the polynomial $x^3 + 3x - 5$ is monic but the polynomial $-x^4 + 2x^3 - 6$ is not.