In each square of a garden shaped like a \(2022 \times 2022\) board, there is initially a tree of height \(0\). A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn:

• The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller.
• The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter.

We say that a tree is majestic if its height is at least \(10^6\). Determine the largest \(K\) such that the gardener can ensure there are eventually \(K\) majestic trees on the board, no matter how the lumberjack plays.