Let x, y, and z be positive real numbers satisfying the system of equations:

\begin{aligned}
& \sqrt{2 x-x y}+\sqrt{2 y-x y}=1 \\
& \sqrt{2 y-y z}+\sqrt{2 z-y z}=\sqrt{2} \\
& \sqrt{2 z-z x}+\sqrt{2 x-z x}=\sqrt{3}
\end{aligned}

Then [(1-x)(1-y)(1-z)]^{2} can be written as \frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.