
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
In the interior of the convex 2011-gon are  $2011$  points, such that no three among the given  $4022$  points (the interior points and the vertices) are collinear. The points are coloured one of two different colours and a colouring is called "good" if some of the points can be joined  in such a way that the following conditions are satisfied:
1) Each segment joins two points of the same colour.
2) None of the line segments intersect.
3) For any two points of the same colour there exists a path of segments connecting them.
Find the number of "good" colourings.

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