ISL 2023 C1

Let $m$ and $n$ be positive integers greater than 1 . In each unit square of an $m \times n$ grid lies a coin with its tail-side up. A move consists of the following steps: 1. select a $2 \times 2$ square in the grid; 2. flip the coins in the top-left and bottom-right unit squares; 3. flip the coin in either the top-right or bottom-left unit square. Determine all pairs $(m, n)$ for which it is possible that every coin shows head-side up after a finite number of moves. (Thailand)