Denote by \(F_0(x), F_1(x), ...\) the sequence of Fibonacci polynomials, which satisfy the recurrence \(F_0(x) = 1, F_1(x) = x,\) and \(F_n(x) = xF_{n−1}(x) + F_{n−2}(x)\) for all \(n ≥ 2\). It is given that there exist unique integers \(λ_0, λ_1, . . ., λ_{1000}\) such that \(x^{1000} = \sum_{i=0}^{1000} λ_iF_i(x)\) for all real \(x\). For which integer \(k\) is \(|λ_k|\) maximized?

Note: Full credit is given to the makers of this problem.

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