
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 angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is given that
$$\frac{3a-b}{\cos B} = \frac{c}{\cos C}$$.  
(1) Find the value of $\sin C$;   
(2) If $c = \sqrt{3}$, find the maximum value of the area of $\triangle ABC$, denoted as $S$.

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