
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 $\triangle ABC$, $BC=a$, $AC=b$, where $a$ 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 measure of angle $C$;
2. The length of $AB$.

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