AT-OEMO-REG 2011 Problem 2

Determine all triples $(x,y,z)$ of real numbers such that the following system of equations holds true: \begin{align*}2^{\sqrt[3]{x^2}}\cdot 4^{\sqrt[3]{y^2}}\cdot 16^{\sqrt[3]{z^2}}&=128 \left(xy^2+z^4\right)^2&=4+\left(xy^2-z^4\right)^2\text{.}\end{align*}