
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 $\triangle ABC$, $b\sin 2A = \sqrt{3}a\sin B$.  
$(Ⅰ)$ Find the value of $\angle A$;  
$(Ⅱ)$ If the area of $\triangle ABC$ is $3\sqrt{3}$, choose one of the following conditions as known to ensure the existence and uniqueness of $\triangle ABC$, and find the value of $a$.  
Condition 1: $\sin C = \frac{2\sqrt{7}}{7}$;  
Condition 2: $\frac{b}{c} = \frac{3\sqrt{3}}{4}$;  
Condition 3: $\cos C = \frac{\sqrt{21}}{7}$  
Note: If the chosen condition does not meet the requirements, no points will be awarded for question $(Ⅱ)$; if multiple suitable conditions are chosen and answered separately, points will be awarded based on the first answer.

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