
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
All segments joining  $n$  points (no three of which are collinear) are coloured in one of  $k$  colours. What is the smallest  $k$  for which there always exists a closed polygonal line with the vertices at some of the  $n$  points, whose sides are all of the same colour?

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