
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 $f(x) = (2\sqrt{3}\cos \omega x + \sin \omega x)\sin \omega x - \sin^2(\frac{\pi}{2} + \omega x), (\omega > 0)$, and the distance from one symmetry center of the graph of the function $y=f(x)$ to the nearest symmetry axis is $\frac{\pi}{4}$.

1. Find the interval(s) where the function $f(x)$ is monotonically increasing.
2. Find the range of the function $f(x)$ on the interval $[0, \frac{\pi}{2}]$.

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