KöMaL 2023 B5341

The centroid of a regular tetrahedron $ABCD$ is $S$, and an arbitrary interior point is $P$. Reflect point $P$ in the planes of the four faces of the tetrahedron to obtain tetrahedron $XYZW$. Show that the centroid of $XYZW$ is the point that divides line segment $PS$ in the ratio $2:1$, with the shorter piece next to $S$.