SE-SKOLORNAS 1986 Problem 4

Prove that $x = y = z = 1$ is the only positive solution of the system $$\left\{ \begin{array}{l} x+y^2 +z^3 = 3 y+z^2 +x^3 = 3 z+x^2 +y^3 = 3 \end{array} \right.$$