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 \(a+b+c.\)

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