Let $n \ge 2$ be a positive integer. There is an $n \times n$ chessboard region, each cell of which is a park. In each park there are some cats (the number of cats is a nonnegative integer). To manage the cats, the administration performs operations on the parks. In each operation, the administration selects a park:
(1) The selected park must have at least as many cats as the number of its neighboring parks.
(2) After selecting park $A$, for each neighboring park $B$ of $A$, the administration moves one cat from $A$ to $B$ (two parks are called neighbors if and only if they share a side).
Let $m$ be the total number of cats in all parks. Find the smallest $m$ for which there exists an initial distribution of cats such that the administration can perform infinitely many operations by suitably choosing the park for each operation.