Denote by F_{0}(x), F_{1}(x), \ldots the sequence of Fibonacci polynomials, which satisfy the recurrence F_{0}(x)=1, F_{1}(x)=x, and F_{n}(x)=x F_{n-1}(x)+F_{n-2}(x) for all n \geq 2 .{ }^{1} It is given that there exist integers \lambda_{0}, \lambda_{1}, \ldots, \lambda_{1000} such that
x^{1000}=\sum_{i=0}^{1000} \lambda_{i} F_{i}(x)
for all real x. For which integer k is \left|\lambda_{k}\right| maximized?