Rotating the roots of a cubic

Algebra Level 5

Consider all pairs of polynomials $(f(x), g(x))$ with complex coefficients such that

1. $f(x)$ is a monic polynomial of degree 3 and has distinct roots $\alpha_1, \alpha_2, \alpha_3$.
2. $g(x)$ is a linear polynomial such that $g(\alpha_1)=\alpha_2,\ g(\alpha_2)=\alpha_3,\ g(\alpha_3)=\alpha_1.$
3. $f(0) = 2$.
4. $f(1) = 9$.

The sum of all possible (distinct) values of $f(2)$ is $N$. What are the last three digits of $N$?

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.

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