
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 $\triangle ABC$, it is known that $\sqrt{3}c=\sqrt{3}bcosA+asinB$.  
$(1)$ Find the value of $B$;  
$(2)$ Choose one of the following three conditions to ensure the unique existence of $\triangle ABC$ and find its area:  
① $b=\sqrt{13},a=4$;  
② $A=\frac{π}{4},b=2\sqrt{3}$;  
③ $c=4,b=\sqrt{21}$.

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