
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 $f(x) = \frac{\sqrt{3}}{2}\sin{2x} - \cos^2{x} - \frac{1}{2}$, ($x \in \mathbb{R}$).
1. When $x \in \left[-\frac{\pi}{12}, \frac{5\pi}{12}\right]$, find the range of the function $f(x)$.
2. Let $\triangle ABC$ have internal angles $A$, $B$, $C$ with corresponding sides $a$, $b$, $c$, and $c= \sqrt{3}$, $f(C)=0$. If the vector $\overrightarrow{m} = (1, \sin{A})$ is collinear with the vector $\overrightarrow{n} = (2, \sin{B})$, find the values of $a$ and $b$.

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