
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 $\left[-\frac{\pi}{4}, \frac{2\pi}{3}\right]$, find the range of possible values for $\omega$.  
(2) For $\omega = 2$, after shifting the graph of the function $y=f(x)$ left by $\frac{\pi}{6}$ units and up by 1 unit, we get the graph of the function $y=g(x)$. If on the interval $[a, b]$ ($a, b \in \mathbb{R}$ and $a < b$), $y=g(x)$ has exactly 30 zeroes, find the range of possible values for $b-a$.

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