
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 a function $f(x) = 2\sin(\omega x + \frac{\pi}{6})$, where $\omega > 0$ is a constant:  
(1) If $y=f(x)$ has at least 10 maximums in the interval $[0, 1]$, find the minimum value of $\omega$.  
(2) Let $\omega=1$, compress the x-coordinates of all the points on the graph of $y=f(x)$ by a factor of $\frac{1}{2}$, and then shift it to the left by $\frac{\pi}{12}$ units to obtain the graph of the function $y=g(x)$. If $g(x)=-1$ has at least 20 solutions in the interval $[m, n]$ ($m, n \in \mathbb{R}$ and $m < n$), find the minimum value of $n-m$ among all intervals satisfying the above condition.

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