
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 $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$, respectively, and $3\cos C = 2\sin A\sin B$.  
$(1)$ Find the minimum value of $\frac{\sin C}{\sin A \sin B}$;  
$(2)$ If $A = \frac{\pi}{6}$ and $a = \sqrt{7}$, find $c$ and the area of $\triangle ABC$.

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