
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 sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The vectors $\overrightarrow{m}=(b,\sqrt{3}a)$ and $\overrightarrow{n}=(\sin B,\sin 2A)$ are given, and $\overrightarrow{m} \parallel \overrightarrow{n}$.  
$(Ⅰ)$ 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: $\cos C = \frac{\sqrt{21}}{7$

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