
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 where the cells are initially colored in a checkerboard pattern, with at least one of the corner cells being black. In one move, you are allowed to repaint the four cells in some \( 2 \times 2 \) square according to the following rule: repaint each white cell to black, each black cell to green, and each green cell to white. For which values of \( n \) is it possible, after several moves, to obtain a checkerboard pattern where the black and white colors have swapped places?

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