There are $2022$ grids in a row. Two people, A and B, play a game with these grids. Initially, each odd-numbered grid from the left is marked with A's name, and each even-numbered grid from the left is marked with B's name. The players take turns performing the following action, starting with player A:

- One should select two grids marked with their own name, such that these two grids are not adjacent, and all grids between them are marked with the opponent's name. Then, change the name in all the grids between those two grids to one's own name.

The game continues until one player can no longer make a move. Find the largest positive integer $m$ such that:

- No matter how player B acts, player A can always ensure that there are at least $m$ grids marked with A's name.

Let's think step by step and output the final answer within \boxed{}.
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