Polynomial Recursion

A sequence of polynomials gk(x)g_k(x) is defined recursively as follows. g0(x)=x; gk+1(x)=gk(x2+2x)gk(x)g_0(x)=x; \ g_{k+1}(x)=g_k(x^2+2x)-g_k(x) Find the last three digits of the coefficient of x2x^2 in g299(x)g_{299}(x).

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