
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 $f(x) = 4 \cos(\omega x - \frac{\pi}{6}) \sin \omega x - \cos(2\omega x + \pi)$, where $\omega > 0$,
(Ⅰ) Find the range of the function $y = f(x)$
(Ⅱ) If $f(x)$ is increasing on 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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