
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 $\triangle ABC$ with the sides opposite to angles $A$, $B$, and $C$ being $a$, $b$, and $c$ respectively, and vectors $\overrightarrow{m}=\left(\cos \frac{C}{2},\sin \frac{C}{2}\right)$, $\overrightarrow{n}=\left(\cos \frac{C}{2},-\sin \frac{C}{2}\right)$, with the angle between $\overrightarrow{m}$ and $\overrightarrow{n}$ being $\frac{\pi}{3}$.

$(1)$ Find the value of angle $C$;

$(2)$ Given $c=3$ and the area of $\triangle ABC$, $S= \frac{4 \sqrt{3}}{3}$, find the value of $a+b$.

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