NZ-NZMO 2023 Problem 4

Let $p$ be a prime and let $f(x) = ax^{2} + bx + c$ be a quadratic polynomial with integer coefficients such that $0 < a, b, c \leq p$ . Suppose $f(x)$ is divisible by $p$ whenever $x$ is a positive integer. Find all possible values of $a + b + c$ .