
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
Given a triangle $∆ABC$, where the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, and it is known that $2\cos C(a\cos B+b\cos A)=c$.

(1) Find angle $C$;

(2) If $c=3$, find the maximum area of $∆ABC$.

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