
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 the axis of symmetry 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 the internal angles $A$, $B$, $C$ of a triangle have corresponding sides $a$, $b$, $c$, respectively, with $c=\sqrt{3}$, and $f(C)=1$. 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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