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Let $F_n$ be the $n$th Fibonacci number, where $F_1=1$, $F_2=1$ and $F_{n+1}=F_n+F_{n-1}$, for $n \geq 2$. Evaluate the sum $\sum_{n=2}^\infty \frac{F_n}{F_{n+1} F_{n-1}}.$

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