2022 AIME I Problem 15

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.