No Fibbing 4

Gx=n=1xFnFn+1Fn+2G_x = \sum_{n=1}^x F_n F_{n+1} F_{n+2}

If GxG_x is defined as above, where FnF_n denotes the nnth Fibonacci number with F1=F2=1F_1 = F_2 = 1, then

x=1GxGx+2Gx+12=a+bc\prod_{x=1}^{\infty} \frac {G_x G_{x+2}}{G_{x+1}^2} = \frac {a + \sqrt{b}}{c}

where aa, bb, and cc are positive integers with bb being square-free. What is a+b+ca + b + c?

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