
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
Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite angles $A$, $B$, and $C$, respectively. Given that $c=2\sqrt{3}$ and $\sin B=2\sin A$.

$(1)$ If $C= \frac{\pi}{3}$, find the values of $a$ and $b$;  
$(2)$ If $\cos C= \frac{1}{4}$, find the area of $\triangle ABC$.

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