Polynomial Inequalities

Algebra Level 5

Suppose $f(x)$ and $g(x)$ are non-constant polynomials with integer coefficients, such that $f$ is monic and $f(g(x))=(f(x))^2\cdot g(x).$ Suppose $N$ is the number of possible polynomials $g,$ such that all coefficients of $g$ have absolute value strictly less than $10^{100}$ . Find 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.