
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
From a  $n\times (n-1)$  rectangle divided into unit squares, we cut the *corner*, which consists of the first row and the first column. (that is, the corner has  $2n-2$  unit squares). For the following, when we say *corner* we reffer to the above definition, along with rotations and symmetry. Consider an infinite lattice of unit squares. We will color the squares with  $k$  colors, such that for any corner, the squares in that corner are coloured differently (that means that there are no squares coloured with the same colour). Find out the minimum of  $k$ .

*Proposed by S. Berlov*

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