
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  $k$  is said  $n$ -squared if by colouring the squares of a  $2n \times k$  chessboard, in any manner, with  $n$  different colours, we can find  $4$  separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in function of  $n$ , the smallest natural  $k$  that is  $n$ -squared.

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