BALTIC-WAY 2003 Problem 3

Let $x, y$ and $z$ be positive real numbers such that $x y z=1$. Prove that

$$(1+x)(1+y)(1+z) \geq 2\left(1+\sqrt[3]{\frac{y}{x}}+\sqrt[3]{\frac{z}{y}}+\sqrt[3]{\frac{x}{z}}\right)$$