12.5.6 Olympic algebra

condition: Solve in positive integers \( x, y, z \) the system of equations: \[ \left\{\begin{array}{l} \frac{1}{x}-20 y+15 z=18 \\ \frac{1}{y}-15 z+18 x=20 \\ \frac{1}{z}-18 x+20 y=15 \end{array}\right. \]