Mysterious 100 degree Monic Polynomials

Algebra Level 5

Suppose \(f(x)\) and \(g(x)\) are coprime monic polynomials with complex coefficients, such that \(f(x)\) divides \(g(x)^2-x\) and \(g(x)\) divides \(f(x)^2-x\). For all such polynomials \(f\) of degree \(100\), what is the largest possible value of \(f(4)\)?

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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