
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 $ABC$ with the sides $a$, $b$, $c$ facing angles $A$, $B$, $C$ respectively. Let $\overrightarrow{m}=(1,1)$ and $\overrightarrow{n}=\left(\frac{\sqrt{3}}{2}-\sin B\sin C,\cos B\cos C\right)$, and $\overrightarrow{m}\perp\overrightarrow{n}$.
1. Determine the magnitude of angle $A$.
2. If $a=1$ and $b=\sqrt{3}c$, find the area of $S_{\triangle ABC}$.

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