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The Fibonacci sequence is defined by $F_0 = 0,,F_1=1$ and $F_n = F_{n - 1} + F_{n - 2}$ for all $n \ge 2.$

Compute $\det \begin{pmatrix} F_{1000} & F_{1001} & F_{1002} \\ F_{1001} & F_{1002} & F_{1003} \\ F_{1002} & F_{1003} & F_{1004} \end{pmatrix} .$