
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
A natural number $n \geq 2$ is given. 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. What is the smallest $k$ for which there will always be four cells painted in four different colors, situated at the intersections of two rows and two columns in any such coloring?

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