
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 $ABC$, $\mathbf{m}=\left( \cos \frac{C}{2},\sin \frac{C}{2} \right)$, $\mathbf{n}=\left( \cos \frac{C}{2},-\sin \frac{C}{2} \right)$, and the angle between $\mathbf{m}$ and $\mathbf{n}$ is $\frac{\pi }{3}$.

$(1)$ Find the angle $C$.

$(2)$ Given that side $C=\frac{7}{2}$ and the area of the triangle $s=\frac{3\sqrt{3}}{2}$, find $a+b$.

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