Solve 3 exponential equations $z^x=x$, $z^y=y$, $y^y=x$ to get $x$, $y$, $z$. The main question is :
$z^x=x$, $z^y=y$, $y^y=x$
Find $z$, $y$, $x$.
My method : 
I first attempted to get two equation for the unknowns $x$ and $y$.
We can happily write :
$z=x^{1/x}$ and $z=y^{1/y}$
Thus we get,
$x^{1/x}=y^{1/y}$
Which is,
$x^y=y^x$.
I can't go any farther than this. Please help me.

$$z^{y}=y$$
$$(z^y)^y=y^y$$
$$z^{y^2}=x=z^x$$
therefore
$$x=y^2$$
on the other hand
$$y^y=x=y^2\implies y=2$$
thus
$$x=4\quad,\quad z=\sqrt{2}$$
