JBMO-SL 2008 A4

Find all triples $(x, y, z)$ of real numbers that satisfy the system

$$\left\{\begin{array}{l} x+y+z=2008 \\ x^{2}+y^{2}+z^{2}=6024^{2} \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2008} \end{array}\right.$$