
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 $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The values of $a$ and $b$ are the roots of the equation $x^{2}-2 \sqrt {3}x+2=0$, and $\cos (A+B)= \frac {1}{2}$.  
$(1)$ Find the degree of angle $C$;  
$(2)$ Find the length of $AB$;  
$(3)$ Find the area of $\triangle ABC$.

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