
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 $f(x) = A\sin(\omega x + \varphi)$ ($A > 0$, $\omega > 0$, $|\varphi| < \pi$), within the same period, when $x= \frac{\pi}{12}$, $f(x)$ reaches its maximum value of 3; when $x= \frac{7\pi}{12}$, $f(x)$ reaches its minimum value of -3.  
(I) Find the expression for the function $f(x)$.  
(II) If $x \in \left[-\frac{\pi}{3}, \frac{\pi}{6}\right]$ and the function $h(x) = 2f(x) + 1 - m$ has two zeros, find the range of the real number $m$.

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