2023 AIME I Problems/Problem 9

Problem (Unofficial, please update when official one comes out):

$P(x) = x^3 + ax^2 + bx + c$ is a polynomial with integer coefficients between $-20$ and $20$ , inclusive. There is exactly one integer $m$ such that $P(m) = P(2)$ . How many possible values are there for the ordered triple $(a, b, c)$ ?