Mixed system of equations and inequation

Algebra Level 4

\[ \begin{cases} 2x-2y+\sqrt { 2x+y+2xy+1 } =1 \\ \sqrt [ 3 ]{ 3y+1 } =8{ x }^{ 3 }-2y-1 \end{cases}\]

If \(x>0\), find the number of pairs of real numbers \((x,y) \) satisfying the equation above.

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