
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
Choose one of the following conditions as the known condition and fill in the blank:①$csinB=bcos(C-\frac{π}{6})$；②$cosB=\frac{2a-b}{2c}$；③$({a}^{2}+{b}^{2}-{c}^{2})tanC=\sqrt{3}ab$. Given that in triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, satisfying _____. Note: If multiple conditions are chosen and solved separately, the first one will be scored.  
$(1)$ Find the value of $\sin C$;  
$(2)$ If $c=\sqrt{3}$ and $a=3b$, find the area of $\triangle ABC$.

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