
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 $\angle A$, $\angle B$, $\angle C$ are $a$, $b$, $c$ respectively, and they satisfy $4\cos C + \cos 2C = 4\cos C \cos^2 \frac{C}{2}$.
(1) Find the measure of $\angle C$;
(2) If $| \overrightarrow{CA} - \frac{1}{2} \overrightarrow{CB} | = 2$, find the maximum area of $\triangle ABC$.

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