
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
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $a+2b=c\cos B+\sqrt{3}c\sin B$, the angle bisector of angle $C$ intersects $AB$ at point $D$ with $BD=\frac{35}{4}$ and $AD=\frac{21}{4}$. Find the measure of $\angle ACB$ and the area of the incircle of $\triangle ABC$.

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