
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 $\triangle ABC$, $BC=a$, $AC=b$, $a$ and $b$ are the two roots of the equation $x^{2}-2 \sqrt {3}x+2=0$, and $2\cos (A+B)=1$. Find:  
$(1)$ The degree measure of angle $C$;  
$(2)$ The length of side $AB$.

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