
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 $\triangle ABC$ with its internal angles $A$, $B$, $C$ opposite to sides $a$, $b$, $c$ respectively, and it satisfies $\frac{\sqrt{3}c}{\cos C}=\frac{a}{\cos (\frac{3\pi }{2}+A)}$.

(I) Find the value of $C$;

(II) If $\frac{c}{a}=2$ and $b=4 \sqrt{3}$, find the area of $\triangle ABC$.

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