
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 a natural number \( n \geq 2 \). Consider all colorings of the cells of an \( n \times n \) board in \( k \) colors such that each cell is painted in exactly one color, and all \( k \) colors are used. For the smallest \( k \), in any such coloring, can there be found four cells painted in four different colors located at the intersections of two rows and two columns?

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