
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-angled 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 \(P\). 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 7 and 5, respectively.

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

b) Suppose it is additionally known that \(\angle ABC = \operatorname{arctg} \frac{3}{4}\). Find \(AC\).

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