
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 the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, $c$, with $a=\sqrt{6}$, $\sin ^{2}B+\sin ^{2}C=\sin ^{2}A+\frac{2\sqrt{3}}{3}\sin A\sin B\sin C$. Choose one of the following conditions to determine whether triangle $\triangle ABC$ exists. If it exists, find the area of $\triangle ABC$; if it does not exist, please explain the reason.  
① The length of the median $AD$ of side $BC$ is $\frac{\sqrt{10}}{2}$  
② $b+c=2\sqrt{3}$  
③ $\cos B=-\frac{3}{5}$.

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