
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
Let \( n \) be a positive integer. Find the smallest positive integer \( k \) with the property that for any coloring of the squares of a \( 2n \times k \) chessboard with \( n \) colors, there are 2 columns and 2 rows such that the 4 squares at their intersections all have the same color.

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