
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 $O$. A circle passing through points $A$, $O$, and $C$ intersects segment $BC$ at point $P$. Tangents to $\omega$ passing through 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 10 and 8 respectively.

a) Find the area of triangle $ABC$.

b) Assume additionally that $\angle ABC = \operatorname{arctg} \frac{1}{2}$. Find $AC$.

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