IMO 1995 — Shortlisted problems Question 4

A4 (USA). Let $a$, $b$, and $c$ be given positive real numbers. Determine all positive real numbers $x$, $y$, and $z$ such that
$$x+y+z=a+b+c$$
and
$$4xyz-(a^2x+b^2y+c^2z)=abc.$$