Real numbers $x, y$, and $z$ are chosen from the interval $[−1, 1]$ independently and uniformly at random. What is the probability that $$\vert x\vert +\vert y\vert +\vert z\vert +\vert x+y+z\vert=\vert x+y\vert +\vert x+z\vert +\vert y+z\vert$$
Now if all of $x, y, z$ are positive or all negative then the equation is of course satisfied. Hence if we consider a 3D space , it denotes two unit cubes, one in the first octant centred at $\left( \frac 12,\frac 12,\frac 12\right)$ and the other in seventh octant centred at $\left( -\frac 12,-\frac 12,-\frac 12\right)$.
The total measure of universal set is the cube with edge length $2$ centred at origin.
But now I have a problem about what if any two of $x, y, z$ are positive while the other remaining be negative or the other way around. Even if I try to make cases it seems to be quite a cumbersome task to approach since we will also need to check signs of $\vert x+y\vert$ and similarly others as well as that of $\vert x+y+z\vert $
I also thought to give a shot using vectors but didn't reach any specific result.
Any help would be quite beneficial.
I would also be happy to see a geometrical intuitive way to attack the problem.