Unlucky Polynomial

Algebra Level 5

Suppose \(f(x)\) is a polynomial of degree \(13\) that satisfies the functional equation \[f(2x^2-1)=2(f(x))^2-1\] Find the last three digits of the coefficient of \(x^5\) in \(f\).

Details and assumptions

The coefficient of \(x^5\) is a positive integer.

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