
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 a fixed circle with radius 1, and let \(BC\) be a fixed chord of \(\omega\) such that \(BC = 1\). The locus of the incenter of \(ABC\) as \(A\) varies along the circumference of \(\omega\) bounds a region \(\mathcal{R}\) in the plane. Find the area of \(\mathcal{R}\).

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