
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 acute triangle $\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, $c$, respectively. The vectors $\overrightarrow{a}=(2a, \sqrt{3})$, $\overrightarrow{b}=(c, \sin C)$, and $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$.  
$(1)$ Find angle $A$.  
$(2)$ If $c=2$ and the area of $\triangle ABC$ is $\frac{3\sqrt{3}}{2}$, find the length of the median $BM$ on side $AC$.

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