
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
The segments \( AB \) and \( BC \) are connected at point \( B \) such that point \( A \) is fixed. Points \( B \) and \( C \) can be in any position in the plane, provided that the three points are not collinear. What is the locus of the center of the circles determined by the three points?

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