Joel's Problem 5

Suppose that in a set $A$ of mutually coprime positive integers, for any ordered triple $(x, y, z)$ in $A$ with $x, y, z$ mutually distinct, $x^{2}$ is a factor of $y^{3}+z^{3}$ . Let $X$ be the number of distinct elements in $A$ . Find the maximum possible value of $X$ .

(If $X<3$ , $(x, y, z)$ might not exist.)