
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 $\triangle ABC$, $BC=a$, $AC=b$, and $a$, $b$ are the roots of the equation $x^{2}-2 \sqrt{3}x+2=0$, $2\cos (A+B)=1$

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

$(2)$ Find the length of $AB$.

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