Question 176
Find all ordered triples $(x, y, z)$ of real numbers such that
$$
\begin{cases} 5\left(x+\frac{1}{x}\right) = 12\left(y+\frac{1}{y}\right) = 13\left(z+\frac{1}{z}\right), \\ xy + yz + zx = 1. \end{cases} \quad \text{(posed by Zhu Huawei)}
$$
$$
\begin{cases} 5\left(x+\frac{1}{x}\right) = 12\left(y+\frac{1}{y}\right) = 13\left(z+\frac{1}{z}\right), \\ xy + yz + zx = 1. \end{cases} \quad \text{(posed by Zhu Huawei)}
$$