If Only It Was (n-2)(n+2)

The Fibonacci sequence is given by \( F_1 = 1\), \(F_2 = 1\), and \( F_{n+1} = F_n + F_{n-1} \). What is \[ \sum_{n=3}^\infty \frac{ 1 } { F_{n-2} F_{n+2} } ?\] If your answer can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).


Inspiration.

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