
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 $\triangle ABC$, $a$ and $b$ are the two roots of the equation $x^2 - 2\sqrt{3}x + 2 = 0$, and $2\cos(A + B) = -1$.
1. Determine the measure of angle $C$.
2. Find the length of side $c$.
3. Calculate the area of $\triangle ABC$.

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