
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 $f(x) = \sqrt{3}\sin\omega x\cos\omega x - \cos^2\omega x + \frac{1}{2}$ ($\omega > 0$), and the axis of symmetry of the graph of $f(x)$ is $x = \frac{\pi}{3}$. The zero of $f(x)$ adjacent to $x = \frac{\pi}{3}$ is $x = \frac{\pi}{12}$.

(Ⅰ) Discuss the monotonicity of the function $f(x)$ in the interval $\left[-\frac{\pi}{12}, \frac{5\pi}{12}\right]$.

(Ⅱ) Let $\triangle ABC$ have internal angles $A$, $B$, $C$ with corresponding sides $a$, $b$, $c$, respectively, and $c = \sqrt{3}$, $f(C) = 1$. If vector $\overrightarrow{m} = (1, \sin A)$ is collinear with 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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