Must They All Be Equal To Each Other?

Algebra Level 2

x=2z21+z2,y=2x21+x2,z=2y21+y2\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)(0,0,0) , find the number of triplets of real numbers (x,y,z)(x,y,z) that satisfy the system of equations above.

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