Holy Fibonacci!

Calculus Level 5

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

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

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