Question 1
Determine all triples $(x, y, z)$ of real numbers satisfying the following system of equations:
$$
\begin{aligned}
2^{\sqrt[3]{x^2}} \cdot 4^{\sqrt[3]{y^2}} \cdot 16^{\sqrt[3]{z^2}} &= 128 \\
(xy^2 + z^4)^2 &= 4 + (xy^2 - z^4)^2.
\end{aligned}
$$
$$
\begin{aligned}
2^{\sqrt[3]{x^2}} \cdot 4^{\sqrt[3]{y^2}} \cdot 16^{\sqrt[3]{z^2}} &= 128 \\
(xy^2 + z^4)^2 &= 4 + (xy^2 - z^4)^2.
\end{aligned}
$$