
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 $A$, $B$, and $C$ are the three internal angles of $\triangle ABC$, and their respective opposite sides are $a$, $b$, and $c$. If $m=\left(-\cos \frac{A}{2}, \sin \frac{A}{2}\right)$, $n=\left(\cos \frac{A}{2}, \sin \frac{A}{2}\right)$, and $m \cdot n = \frac{1}{2}$.
(1) Find the measure of angle $A$;
(2) If $a=2\sqrt{3}$, the area of the triangle $S=\sqrt{3}$, find the value of $b+c$.

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