
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 positive integer \( n \geq 3 \), let \( S \) be a finite set of points on a plane with \( |S| = n \), where no three points are collinear and no four points are concyclic. If it is possible to color each point either red or white such that there exists a circle where all the points inside the circle are white and all the points outside are red, the coloring is called "appropriate." How many such appropriate colorings are there?

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