A composition of polynomials

Suppose \(f\) and \(g\) are polynomials with integer coefficients such that \(f(0)=0,\) \(f\) has degree at least two, and \[f(g(x))=f(x)+(x^6+3x^5-6x^3+6x^2).\] Find the last three digits of the sum of absolute values of all distinct possible values of \(f(10)\).

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