
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 \( \triangle ABC \) be a triangle inscribed in a unit circle with center \( O \). Let \( I \) be the incenter of \( \triangle ABC \), and let \( D \) be the intersection of \( BC \) and the angle bisector of \( \angle BAC \). Suppose that the circumcircle of \( \triangle ADO \) intersects \( BC \) again at a point \( E \) such that \( E \) lies on \( IO \). If \( \cos A = \frac{12}{13} \), find the area of \( \triangle ABC \).

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