
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 with \( k \) colors, such that each cell is painted in exactly one color and all \( k \) colors are used. Determine the smallest \( k \) for which, in any such coloring, there will be four cells painted in four different colors located at the intersection of two rows and two columns.

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