CMO 1996 Problem 2

Find all real solutions to the following system of equations. Carefully justify your answer. $$\left\{ \begin{array}{c} \frac{4x^2}{1+4x^2} = y \frac{4y^2}{1+4y^2} = z \frac{4z^2}{1+4z^2} = x \end{array} \right.$$