
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 $ABC$, let the lengths of sides opposite to angles $A$, $B$, and $C$ be $a$, $b$, and $c$ respectively. Given vectors $\vec{m} = (\sin C, \sin B \cos A)$ and $\vec{n} = (b, 2c)$, and that $\vec{m} \cdot \vec{n} = 0$,
(1) Find the magnitude of $\angle A$;
(2) If $a = 2\sqrt{3}$ and $\sin B + \sin C = 1$, find the area $S$ of $\triangle ABC$.

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