
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 colouring nof the squares of a  $2n$  by  $k$  chessboard with  $n$  colours, there are  $2$  columns and  $2$  rows such that the  $4$  squares in their intersections have the same colour.

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