
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
In a grid square of size \( n \times n \), where each cell is painted in one of two colors: white or black, what is the smallest \( n \) for which a rectangle always exists (i.e., for any coloring) whose vertices coincide with the centers of four cells of the same color?

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