Interpolation and Integers

Let \(a,b,c \in \mathbb{Z_4}\) be interpolated by a quadratic polynomial \(f\). That is, \(f(1) = a, f(2) = b\) and \( f(3) =c\). For how many triples of \((a,b,c)\) does \(f\) have integer (including zero) coefficients?

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