
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
The quadrilateral \(ABCD\) is inscribed in a circle with a radius of 7. Rays \(AB\) and \(DC\) intersect at point \(P\), while rays \(BC\) and \(AD\) intersect at point \(Q\). It is known that triangles \(ADP\) and \(QAB\) are similar (the vertices are not necessarily listed in the corresponding order).

a) Find \(AC\).

b) Additionally, it is known that the circles inscribed in triangles \(ABC\) and \(ACD\) touch segment \(AC\) at points \(K\) and \(T\) respectively, and \(CK: KT: TA = 5: 2: 7\) (point \(T\) lies between \(K\) and \(A\)). Find \(\angle DAC\) and the area of the quadrilateral \(ABCD\).

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