
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 $f(x) = 2\sin(\omega x)$, where $\omega > 0$ is a constant,  
(1) If $y = f(x)$ is monotonically increasing on the interval $\left[-\frac{\pi}{4}, \frac{2\pi}{3}\right]$, find the range of values for $\omega$;  
(2) Let $\omega = 2$, and the graph of the function $y = f(x)$ is translated left by $\frac{\pi}{6}$ units and then translated up by $1$ unit to obtain the graph of the function $y = g(x)$. For the interval $[a, b]$ $(a, b \in \mathbb{R}$ and $a < b)$ such that $y = g(x)$ has at least $30$ zeros on $[a, b]$. Among all such intervals $[a, b]$, find the minimum value of $(b - a)$.

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