
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 a triangle $\triangle ABC$ with angles $A$, $B$, and $C$ and their corresponding sides represented by vectors $\overrightarrow{m} = (-\cos \frac{A}{2}, \sin \frac{A}{2})$ and $\overrightarrow{n} = (\cos \frac{A}{2}, \sin \frac{A}{2})$. It is known that $\overrightarrow{m} \cdot \overrightarrow{n} = \frac{1}{2}$.

1. If $\sqrt{2}a = \sqrt{3}b$, find angle $B$.
2. If $a = 2\sqrt{3}$ and the area of $\triangle ABC$ is $\sqrt{3}$, find the perimeter of $\triangle ABC$.

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