Short distance teleportation

Algebra Level 5

\[\begin{cases} \dfrac{1}{\sqrt{1+2x^{2}}}+\dfrac{1}{\sqrt{1+2y^{2}}}=\dfrac{2}{\sqrt{1+2xy}} \\ x\sqrt{x(1-2x)}+\sqrt{y(1-2y)}=\dfrac{119}{3200} \cdot \sqrt{\dfrac{39}{2}} \end{cases} \]

If \(x\) and \(y\) satisfy the system of equations above, find the maximum value of \(x+y\).

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