
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 $ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. The vector $\overrightarrow{m}=(b,-\sqrt{3}a)$ is perpendicular to the vector $\overrightarrow{n}=(\cos A,\sin B)$.  
$(1)$ Find $A$;  
$(2)$ If $B+ \frac{\pi}{12}=A$ and $a=2$, find the area of $\triangle ABC$.

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