# No Fibbing 4

$G_x = \sum_{n=1}^x F_n F_{n+1} F_{n+2}$

If $G_x$ is defined as above, where $F_n$ denotes the $n$th Fibonacci number with $F_1 = F_2 = 1$, then

$\prod_{x=1}^{\infty} \frac {G_x G_{x+2}}{G_{x+1}^2} = \frac {a + \sqrt{b}}{c}$

where $a$, $b$, and $c$ are positive integers with $b$ being square-free. What is $a + b + c$?

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