
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 $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and $\overrightarrow{m}=(\cos \frac{C}{3},\sin \frac{C}{2}), \overrightarrow{n}=(\cos \frac{C}{2},-\sin \frac{C}{2})$. The angle between $\overrightarrow{m}$ and $\overrightarrow{n}$ is $\frac{\pi}{3}$.  
$(1)$ Find $\angle C$;  
$(2)$ Given $c= \frac{7}{2}, {S}_{\triangle ABC}= \frac{3}{2} \sqrt{3}$, find $a+b$.

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