
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
4. Quadrilateral $A B C D$ is inscribed in a circle with a radius of 7. Rays $A B$ and $D C$ intersect at point $P$, and rays $B C$ and $A D$ intersect at point $Q$. It is known that triangles $A D P$ and $Q A B$ are similar (not necessarily in the given order).

a) Find the length of $A C$.

b) Additionally, let it be known that the circles inscribed in triangles $A B C$ and $A C D$ touch the segment $A C$ at points $K$ and $T$, respectively, with the ratio $C K: K T: T A=6: 1: 7$ (point $T$ lies between $K$ and $A). Find the measure of $\angle D A C$ and the area of quadrilateral $A B C D$.

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