ILL 1992 Problem 33

Let $a, b, c$ be positive real numbers and $p, q, r$ complex numbers. Let $S$ be the set of all solutions $(x, y, z)$ in $\mathbb C$ of the system of simultaneous equations $$ax + by + cz = p,$$$$ax2 + by2 + cz2 = q,$$$$ax3 + bx3 + cx3 = r.$$ Prove that $S$ has at most six elements.