
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
Given $\overrightarrow{m}=(2\sqrt{3},1)$, $\overrightarrow{n}=(\cos^2 \frac{A}{2},\sin A)$, where $A$, $B$, and $C$ are the interior angles of $\triangle ABC$;  
$(1)$ When $A= \frac{\pi}{2}$, find the value of $|\overrightarrow{n}|$;  
$(2)$ If $C= \frac{2\pi}{3}$ and $|AB|=3$, when $\overrightarrow{m} \cdot \overrightarrow{n}$ takes the maximum value, find the magnitude of $A$ and the length of side $BC$.

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