
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
Given a finite number of points in the plane such that no three points are collinear, what is the maximum number of points for which the line segments connecting them can be colored with three colors so that the following conditions are satisfied:

a) No point can have three segments of different colors emanating from it;

b) It is not possible to find three points such that the three segments they determine are of the same color;

c) All three colors are used.

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