Consider the sequence of Fibonacci numbers F_{0}, F_{1}, F_{2}, \ldots, given by F_{0}=F_{1}=1 and F_{n+1}= F_{n}+F_{n-1} for n \geq 1. Define the sequence x_{0}, x_{1}, x_{2}, \ldots by x_{0}=1 and x_{k+1}=x_{k}^{2}+F_{2^{k}}^{2} for k \geq 0. Define the sequence y_{0}, y_{1}, y_{2}, \ldots by y_{0}=1 and y_{k+1}=2 x_{k} y_{k}-y_{k}^{2} for k \geq 0. If$\sum_{k=0}^{\infty} \frac{1}{y_{k}}=\frac{a-\sqrt{b}}{c}$for positive integers a, b, c with \operatorname{gcd}(a, c)=1, find a+b+c.