
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
100 unit squares of an infinite squared plane form a  $ 10\times 10$  square.  Unit segments forming these squares are coloured in several colours.  It is known that the border of every square with sides on grid lines contains segments of at most two colours. (Such square is not necessarily contained in the original  $ 10\times 10$  square.) What maximum number of colours may appear in this colouring? 

*Author: S. Berlov*

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