
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
João managed to paint the squares of an \( n \times n \) board in black and white so that the intersections of any two rows and any two columns did not consist of squares with the same color. What is the maximum value of \( n \)?

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