Should I really compute that monster?

Let \(F_n\) denote the \(n^\text{th} \) Fibonacci number.

Compute \(\gcd(F_{2000}F_{16} + F_{1999}F_{15}, F_{1014}F_{1001} + F_{1013}F_{1000}) \).

Notation: \(\gcd(a,b) \) denotes greatest common divisor of \(a\) and \(b\).

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