
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 it is known that $(2a+b)\cos C + c \cos B = 0$.
(1) Find the measure of $\angle C$;
(2) If $c=4$, find the values of $a$ and $b$ that maximize the area of $\triangle ABC$.

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