\[F_n = \sqrt { { F }_{ 2001 }{ F }_{ 1999 }+\frac { { F }_{ 2016 }{ F }_{ 2000 }-{ F }_{ 2010 }{ F }_{ 2006 } }{ 440 } }\]

#### This problem is original.

#### Picture credits: *Aloe polyphylla spiral* by Just chaos, Wikipedia

Let \(F_n\) denote the \(n^\text{th} \) Fibonacci number. Find the value of \(n\) such that the equation above is fulfilled.

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