
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 possible colorings of the cells of an \( n \times n \) board in \( k \) colors, such that every cell is painted exactly one color and all \( k \) colors are used. What is the smallest \( k \) such that 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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