Let $x, y$ and $z$ be three real numbers satisfying the following conditions:
$$0 < x \leq y \leq z$$
$$xy + yz + zx = 3$$
Prove that the maximum value of $(x y^3 z^2)$ is $2.$
I tried using the weighted AM-GM inequality, but to no avail as the powers 1,2 and 3 are giving me a hard time. How should I proceed? Thanks in advance.