Mysterious 100 degree Monic Polynomials

Algebra Level 5

Suppose f(x)f(x) and g(x)g(x) are coprime monic polynomials with complex coefficients, such that f(x)f(x) divides g(x)2xg(x)^2-x and g(x)g(x) divides f(x)2xf(x)^2-x. For all such polynomials ff of degree 100100, what is the largest possible value of f(4)f(4)?

Details and assumptions

A polynomial is monic if its leading coefficient is 1. For example, the polynomial x3+3x5 x^3 + 3x - 5 is monic but the polynomial x4+2x36 -x^4 + 2x^3 - 6 is not.

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