
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 points \( A \) and \( B \). Determine the locus of points \( C \) such that \( C \), the midpoints of segments \( AC \) and \( BC \), and the centroid of the triangle \( ABC \) lie on the same circle.

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