
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 $\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given $6\cos B\cos C-1=3\cos \left(B-C\right)$. Find:<br/>
$(1)$ If $B=\frac{π}{6}$, find $\cos C$;<br/>
$(2)$ If $c=3$, point $D$ lies on side $BC$ and $AD$ bisects $∠BAC$ with $AD=\frac{8\sqrt{3}}{7}$, find the area of $\triangle ABC$.

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