
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 that $a$, $b$, $c$ are the opposite sides of the acute angles $A$, $B$, $C$ of triangle $\triangle ABC$, $\overrightarrow{m}=(3a,3)$, $\overrightarrow{n}=(-2\sin B,b)$, and $\overrightarrow{m} \cdot \overrightarrow{n}=0$.
$(1)$ Find $A$;
$(2)$ If $a=2$ and the perimeter of $\triangle ABC$ is $6$, find the area of $\triangle ABC$.

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