
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) Let constant \(\omega > 0\). If \(y = f(\omega x)\) is an increasing function on the interval \(\left[-\frac{\pi}{2}, \frac{2\pi}{3}\right]\), find the range of values for \(\omega\).

(2) Let the set \(A = \left\{x \mid \frac{\pi}{6} \leqslant x \leqslant \frac{2\pi}{3}\right\}\) and \(B = \{x \mid \left| f(x) - m \right| < 2\}\). If \(A \subseteq B\), find the range of values for the real number \(m\).

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