
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
Given an $n \times n$ square initially colored in a checkerboard pattern where at least one corner cell is black. In one move, you are allowed to repaint all four cells in a $2 \times 2$ sub-square according to the following rule: each white cell becomes black, each black cell becomes green, and each green cell becomes white. For which values of $n$ can you achieve a checkerboard pattern where the black and white colors have swapped places through a series of moves?

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