# 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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