
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. Each cell of an  $n \times n$  table is coloured in one of  $k$  colours where every colour is used at least once. Two different colours  $A$  and  $B$  are said to touch each other, if there exists a cell coloured in  $A$  sharing a side with a cell coloured in  $B$ . The table is coloured in such a way that each colour touches at most  $2$  other colours. What is the maximal value of  $k$  in terms of  $n$ ?
 

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