2022 AIME I Problem 15

Let $x$, $y$, and $z$ be positive real numbers satisfying the system of equations
$\sqrt{2x - xy} + \sqrt{2y - xy} = 1$
$\sqrt{2y - yz} + \sqrt{2z - yz} = \sqrt{2}$
$\sqrt{2z - zx} + \sqrt{2x - zx} = \sqrt{3}.$
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 $m+n$.

Leading zeroes must be inputted, so if your answer is 34, then input 034. Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.