
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 two points \( O \) and \( M \):

a) Determine the set of points on the plane that can serve as one of the vertices of a triangle with the center of the circumscribed circle at point \( O \) and the centroid at point \( M \).

b) Determine the set of points on the plane that can serve as one of the vertices of an obtuse triangle with the center of the circumscribed circle at point \( O \) and the centroid at point \( M \).

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