
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
In triangle \(ABC\), the sides \(AB\) and \(BC\) are equal to \(a\), and the angle \(ABC\) is \(120^\circ\). A circle is inscribed in the triangle, touching the side \(AB\) at point \(D\). A second circle, centered at point \(B\), passes through point \(D\). Find the area of the part of the inscribed circle that lies inside the second circle.

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