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Algebra Level 4

\[ \begin{cases}{x+y=\sqrt{4z-1}} \\ {y+z=\sqrt{4x-1}} \\ {z+x=\sqrt{4y-1}}\end{cases} \]

Find the sum of all values of \(x+y+z\) such that \(x,y,z\) satisfy the system of equations above.

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