Irrational Equations

Algebra Level 5

We are given the following system of equations

$\begin{cases} \frac{1}{\sqrt{1+2x^2}}+\frac{1}{\sqrt{1+2y^2}}=\frac{2}{\sqrt{1+2xy}}, \\ \sqrt{x(1-2x)}+\sqrt{y(1-2y)}=\frac{2}{9}. \end{cases}$

If $x$ can be expressed in the form $\frac{a\pm\sqrt{b}}{c},$ where $a,b$ and $c$ are positive integers and $b$ is not divisible by the square of any prime, find the value of $a+b+c.$