
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 $S=\frac{{\sqrt{3}}}{4}({a^2}+{b^2}-{c^2})$ (1) and $a\cos B+b\cos A=2c\cos C$ (2), please choose one of the two conditions and complete the following questions. In triangle $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, let $S$ be the area of $\triangle ABC$, satisfying _______ (fill in the number).  
(1) Find the measure of angle $C$;  
(2) If $c=3$, find the maximum perimeter of $\triangle ABC$.  
Note: If multiple conditions are chosen and answered separately, the first answer will be scored.

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