\left.\begin{array}{l}{[\text { Systems of exponential equations and inequalities] }} \\ {[\quad \text { Monotonicity and boundedness }}\end{array}\right]

Solve the system in positive numbers:

$$
\begin{cases}x^{y} & =z \\ y^{z} & =x \\ z^{x} & =y\end{cases}
$$
First, note that if one of the unknowns is equal to one, then the others are also equal to one. Indeed,

Answer

$x=y=z=1$.