
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 in triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and the vectors $\overrightarrow{m}=\left(2b-a,c\right)$ and $\overrightarrow{n}=\left(\cos A,\cos C\right)$ are collinear.  
$(1)$ Find the value of angle $C$.  
$(2)$ If $c=\sqrt{3}$ and the area of $\triangle ABC$ is $\frac{{\sqrt{3}}}{2}$, find the value of $a+b$.

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