In each square of a garden shaped like a $2022\times2022$ 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 number $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board,no matter how the lumberjack plays. 

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