
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 \( n \times n \) has its cells initially colored in a checkerboard pattern with black and white colors, with at least one of the corner cells being black. In one move, it is allowed to simultaneously recolor the four cells of a \( 2 \times 2 \) subgrid according to the following rule:

- Each white cell is recolored to black.
- Each black cell is recolored to green.
- Each green cell is recolored to white.

For which values of \( n \) is it possible, after several moves, to obtain a checkerboard coloring where the positions of the black and white cells are swapped?

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