
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 $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $a+b=13$, $c=7$, and $4\sin^2\left(\frac{A+B}{2}\right)-\cos2C=\frac{7}{2}$.

(1) Find the magnitude of angle $C$.

(2) Calculate the area of the inscribed circle of triangle $ABC$.

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