They Are All Related To Each Other!

Algebra Level 3

{x+y=4z−1y+z=4x−1x+z=4y−1\large \begin {cases} x+y=\sqrt{4z-1}\\y+z=\sqrt{4x-1}\\x+z=\sqrt{4y-1}\end {cases}

How many ordered triplets of real numbers (x,y,z)(x,y,z) satisfy the system of equations above?

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