Question 123
Find all threes $(x, y, z)$ of real (but not necessarily positive) numbers that meet $$
\begin{aligned}
3\left(x^{2}+y^{2}+z^{2}\right) & =1 \\
x^{2} y^{2}+y^{2} z^{2}+z^{2} x^{2} & =x y z(x+y+z)^{3} .
\end{aligned}
$$
\begin{aligned}
3\left(x^{2}+y^{2}+z^{2}\right) & =1 \\
x^{2} y^{2}+y^{2} z^{2}+z^{2} x^{2} & =x y z(x+y+z)^{3} .
\end{aligned}
$$