
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 \(\omega\) be the incircle of a fixed equilateral triangle \(ABC\). Let \(\ell\) be a variable line that is tangent to \(\omega\) and meets the interior of segments \(BC\) and \(CA\) at points \(P\) and \(Q\), respectively. A point \(R\) is chosen such that \(PR=PA\) and \(QR=QB\). Find all possible locations of the point \(R\), over all choices of \(\ell\).

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