MathNet 2023-part-06 — 191 problems (EN–VI) Question 110

Let $n$ and $m$ be positive integers. A kindergarten teacher arranges $n \times m$ square foam mats into an $n \times m$ rectangular area. On each mat there is a baby, each facing one of the sides of its square. When the teacher claps once, all babies simultaneously move forward one mat in the direction they are facing, and then all babies turn $90$ degrees clockwise in place. If a baby crawls out of the $n \times m$ rectangular area, that baby cries. If two babies move onto the same mat, they collide and both cry.
It is known that the teacher can choose the initial facing direction of each baby so that no baby ever cries, no matter how many times the teacher claps.
Find all possible values of $n$ and $m$.