
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 \( f(x) = 4 \sin x \cdot \sin^{2}\left(\frac{\pi}{4} + \frac{x}{2}\right) + \cos 2x \):
1. Suppose \( \omega > 0 \) is a constant. If the function \( y = f(\omega x) \) is increasing on the interval \(\left[-\frac{\pi}{2}, \frac{2\pi}{3}\right]\), find the range of values for \(\omega\).
2. Let the set \( A = \left\{ x \left|\, \frac{\pi}{6} \leqslant x \leqslant \frac{2\pi}{3} \right. \right\} \) and \( B = \left\{ x \big|| f(x) - m \big| < 2 \right\} \). If \( A \cup B = B \), determine the range of the real number \( m \).

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