
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 $f(x) = \frac{\sqrt{3}}{2} - \sqrt{3} \sin^{2}{\omega x} - \sin{\omega x}\cos{\omega x} (\omega > 0)$, and the distance from one symmetry center of the graph of $y=f(x)$ to the nearest symmetry axis is $\frac{\pi}{4}$.
(I) Find the monotonic intervals of $f(x)$ on $\left[-\frac{\pi}{2}, 0\right]$;
(II) If $f(x_{0}) = \frac{3}{5}$, and $x_{0} \in \left[0, \frac{\pi}{3}\right]$, find the value of $\sin 2x_{0}$.

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