\[\large \sum_{n=0}^\infty \frac{1}{F_{2n+1}+1}=\frac{\sqrt{A}}{B}\]

Let \(F_n\) denote the \(n^\text{th} \) Fibonacci number, where \(F_0 = 0, F_1 = 1\) and \(F_n = F_{n-1} + F_{n-2} \) for \(n=2,3,4,\ldots \).

If the equation above holds true for positive integers \(A\) and \(B\), with \(A\) square-free, find \(A+B\).

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