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

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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