Must They All Be Equal To Each Other?

Algebra Level 4

\[\large x=\frac{2z^2}{1+z^2}, \quad y=\frac{2x^2}{1+x^2}, \quad z=\frac{2y^2}{1+y^2} \]

Excluding the trivial solution \((0,0,0) \), find the number of triplets of real numbers \((x,y,z) \) that satisfy the system of equations above.

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