
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
Let \(ABC\) be a triangle that satisfies \(AB=13, BC=14, AC=15\). Given a point \(P\) in the plane, let \(P_{A}, P_{B}, P_{C}\) be the reflections of \(A, B, C\) across \(P\). Call \(P\) good if the circumcircle of \(P_{A}P_{B}P_{C}\) intersects the circumcircle of \(ABC\) at exactly one point. The locus of good points \(P\) encloses a region \(\mathcal{S}\). Find the area of \(\mathcal{S}\).

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