Strange System of Equations

Algebra Level 5

\(x\) and \(y\) are real numbers satisfying the system of equations \[\left\{\begin{array}{l}y^2+2y=x^2-2x+\sqrt{x}\\ x+y=\dfrac{1}{2\sqrt{x}}\end{array}\right.\]

If the value of \(xy\) can be expressed as \[\dfrac{a+\sqrt{b}}{c}\] for integers \(a,b,c\) with \(b\) square-free, then what is \(a+b+c\)?

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