\[\prod_{n=2}^{\infty} \left(1-\dfrac{2}{F_{n+1}^{2}-F_{n-1}^{2}+1}\right) = \ ? \]

###### Image Credit: Flickr Sam Felder.

Let \(F_n\) denote the \(n^\text{th}\) term of the Fibonacci Sequence. Evaluate the product above.

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