
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 $f(x) = 3\sin(2\omega x + \frac{\pi}{3})$, where $0 < \omega < 2$, if the point $(-\frac{\pi}{6}, 0)$ is a center of symmetry for the graph of function $f(x)$, (1) find the value of $\omega$; (2) determine the period of $f(x)$ and its intervals of monotonic increase; (3) find the solution set for $f(x) \geq \frac{3}{2}$.

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