
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 a circle with center $O$ and a point $P$ outside the circle (but in the same plane), we draw an arbitrary diameter of the circle with endpoints $A$ and $B$. We then connect $A$ and $B$ to $P$. A circle is drawn through points $A$, $B$, and $P$. Determine the locus of the center of this circle as diameter $AB$ rotates around $O$.

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