
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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.

Present the answer in LaTex format: \boxed{Your answer}
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