
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 5 points on a plane where no 3 points are collinear and no 4 points are concyclic. A circle passing through any 3 of these points with one of the remaining two points inside the circle and the other outside the circle is called a "good circle." Find all possible values of the number of good circles, $n$.

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