Madhava from Sangamagrama?

Calculus Level 5

Let \(\{F_n\}\) to be the Fibonacci's sequence, where \(F_1 = F_2=1\) and \(F_{n + 1} = F_n + F_{n - 1}, \space \forall \, n \ge 2\), and \[\displaystyle \sum_{n = 1}^\infty \arctan \left(\frac{1}{F_{2n + 1}}\right) = \frac{A\pi}{B},\] where \(A\) and \(B\) are coprime positive integers. Find \(10A + B\).

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