
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
Five points are marked on a sphere, no three of which lie on a great circle (a great circle is the intersection of the sphere with a plane passing through its center). Two great circles that do not pass through any of the marked points are called equivalent if one can be continuously moved into the other on the sphere without passing through any of the marked points during the process.

a) How many great circles can be drawn that do not pass through the marked points and are not equivalent to each other?

b) The same problem for $n$ marked points.

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