
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 rays \( AB \) and \( DC \) intersect at point \( P \), and the rays \( BC \) and \( AD \) intersect at point \( Q \). It is known that the triangles \( ADP \) and \( QAB \) are similar (vertices are not necessarily listed in corresponding order), and the quadrilateral \( ABCD \) can be inscribed in a circle with radius 4.

a) Find \( AC \).

b) Additionally, it is known that the circles inscribed in triangles \( ABC \) and \( ACD \) touch the segment \( AC \) at points \( K \) and \( T \) respectively, and \( CK: KT: TA = 3:1:4 \) (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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