CGMO 2005 Problem 2

Find all ordered triples $(x, y, z)$ of real numbers such that

$$5 \left(x + \frac{1}{x} \right) = 12 \left(y + \frac{1}{y} \right) = 13 \left(z + \frac{1}{z} \right),$$

and $$xy + yz + zy = 1.$$