DE-MO 2019 Problem 6

Suppose that real numbers $x,y$ and $z$ satisfy the following equations:

\begin{align*} x+\frac{y}{z} &=2, y+\frac{z}{x} &=2, z+\frac{x}{y} &=2. \end{align*}

Show that $s=x+y+z$ must be equal to $3$ or $7$ .

Note: It is not required to show the existence of such numbers $x,y,z$ .