
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 $f(x) = 4 \cos(\omega x - \frac{\pi}{6})\sin(\pi - \omega x) - \sin(2\omega x - \frac{\pi}{2})$, where $\omega > 0$.
(1) Find the range of values for the function $f(x)$.
(2) If $y = f(x)$ is increasing in the interval $\left[-\frac{3\pi}{2}, \frac{\pi}{2}\right]$, find the maximum value of $\omega$.

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