Not-So-Dumb Fibonacci Sum #1

Find the value of \[ \sum_{n=1}^{\infty} \dfrac{F_{n+1}}{F_{n}F_{n+2}} \]

where \(F_n\) is the \(n\)th Fibonacci number, defined by the recurrence relation \(F_n=F_{n-1}+F_{n-2}\) with \(F_1=F_2=1\).

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