
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 \( f(x) = 4 \sin x \cdot \sin^2\left(\frac{\pi}{4} + \frac{x}{2}\right) + \cos 2x \), and a constant \( w > 0 \). If \( y = f(wx) \) is a monotonically increasing function on the interval \(\left[-\frac{\pi}{2}, \frac{2\pi}{3}\right]\), then the range of values for \( w \) is __.

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