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Algebra Level 4

{x+y=4z1y+z=4x1z+x=4y1 \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+zx+y+z such that x,y,zx,y,z satisfy the system of equations above.

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