
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
Determine all values of  $n$  for which there is a set  $S$  with  $n$  points, with no 3 collinear, with the following property: it is possible to paint all points of  $S$  in such a way that all angles determined by three points in  $S$ , all of the same color or of three different colors, aren't obtuse. The number of colors available is unlimited.

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