
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
A square grid of size \(n \times n\) is initially colored in a black and white checkerboard pattern, with at least one of the corner cells being black. In one move, it is allowed to recolor all four cells in a \(2 \times 2\) square according to the following rule: each white cell is recolored black, each black cell is recolored green, and each green cell is recolored white. For which \(n\) is it possible, after a number of moves, to achieve a checkerboard coloring where the black and white colors have swapped places?

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