
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
The game of tri-chess is played on a triangular grid. The grid is an equilateral triangle with side length $n$ and is composed of $n^2$ equilateral triangles with side length 1. In this game, pieces are situated at the vertices of the triangles. Two rooks threaten each other if the line connecting their positions is parallel to one of the sides of the grid. For example, below, the set of vertices threatened by rook A is represented in red; thus, rook A threatens rook B but not rook C.

When $n=2022$, what is the maximum number of rooks that can be placed on the grid such that no two rooks threaten each other?

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