
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 locus of all the centers of circles of a given radius \( a \), in the same plane, passing through two fixed points \( A \) and \( B \), is:
**A)** The midpoint of \( AB \).
**B)** A segment of the perpendicular bisector of \( AB \).
**C)** A complete circle.
**D)** No locus exists.

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