
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 the constant $\omega > 0$,
(1) If $y = f(x)$ is monotonically increasing on the interval $\left[-\frac{3\pi}{4}, \frac{\pi}{3}\right]$, find the range of $\omega$;
(2) Let $\omega = 2$, shift the graph of the function $y = f(x)$ to the left by $\frac{\pi}{6}$ units and then shift upward by 1 unit to obtain the graph of the function $y = g(x)$. For the interval $[a, b]$ ($a < b$, $a, b \in \mathbb{R}$) to satisfy: $y = g(x)$ has at least 20 zeros on $[a, b]$. Among all such intervals $[a, b]$ that satisfy the condition, find the minimum value of $b-a$.

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