Sum of Odd-Indexed Fibonacci

Let $F_n$ be the $n$th Fibonacci number such that $F_1 = F_2 = 1$. Find the exact value of: $$\prod_{n=2}^\infty\left(1+\frac{1}{\left(F_{2n-1}\right)^2}\right)$$The answer can be expressed as $\displaystyle \frac{a+\sqrt{b}}{c}$ where $a$, $b$, $c$ are positive integers, $\gcd(a,c) = 1$, and $b$ is square free. Find the value of $a+b+c$.