
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
Each cell of  $2024\times 2024$  table colored either red, bule, or white. We place a red pawn on each red cell, and a blue pawn on each blue cell. Moreover, for each white cell that shares an edge or vertex with one or more blue cells, place a blue pawn on it. Suppose that for any  $2\times2$  square, the number of red pawns and blue pawns placed within it is equal and either one or two. Find the maximum possible value for the number of squares painted white.

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