
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
An acute triangle \( ABC \) is inscribed in a circle \(\omega\) with center \( O \). The circle passing through points \( A \), \( O \), and \( C \) intersects segment \( BC \) at point \( P \). The tangents to \(\omega\) drawn at points \( A \) and \( C \) intersect at point \( T \). Segment \( TP \) intersects side \( AC \) at point \( K \). It is known that the areas of triangles \( APK \) and \( CPK \) are 12 and 9, respectively.

a) Find the area of triangle \( ABC \).

b) Additionally, suppose that \(\angle ABC = \arctan \frac{3}{7}\). Find \( AC \).

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