
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
Given the function $f(x) = \frac{\sqrt{3}}{2}\sin{2x} - \cos^2{x} - \frac{1}{2}$ ($x \in \mathbb{R}$).
(I) Find the intervals of increase for the function $f(x)$.
(II) Let $\triangle ABC$ have internal angles $A$, $B$, $C$ at vertices $A$, $B$, $C$, respectively, with opposite sides $a$, $b$, $c$, and $c = \sqrt{3}$. Given $f(C) = 0$, and vectors $\overrightarrow{m} = (1, \sin{A})$ and $\overrightarrow{n} = (2, \sin{B})$ are collinear, find the values of $a$ and $b$.

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