
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
Let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, respectively. Let $\overrightarrow{m}=(\cos \frac{C}{2}, \sin \frac{C}{2})$, $\overrightarrow{n}=(\cos \frac{C}{2}, -\sin \frac{C}{2})$, and the angle between $\overrightarrow{m}$ and $\overrightarrow{n}$ be $\frac{\pi}{3}$.  
(1) Find the magnitude of angle $C$.  
(2) Given $c= \frac{7}{2}$ and the area of $\triangle ABC$, $S= \frac{3 \sqrt{3}}{2}$, find the value of $a+b$.

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