Let x, y be distinct positive real numbers satisfying
\frac{1}{\sqrt{x+y}-\sqrt{x-y}}+\frac{1}{\sqrt{x+y}+\sqrt{x-y}}=\frac{x}{\sqrt{y^{3}}}
If \frac{x}{y}=\frac{a+\sqrt{b}}{c} for positive integers a, b, c with \operatorname{gcd}(a, c)=1, find a+b+c.