
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 vectors $\overrightarrow{a}=(-\cos 2x, 2)$ and $\overrightarrow{b}=(2, 2-\sqrt{3} \sin 2x)$, and the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} - 4$.
(I) If $x \in [0, \frac{\pi}{2}]$, find the maximum value of $f(x)$ and the corresponding value of $x$;
(II) If the vertical coordinates of all points on the graph of $f(x)$ are reduced to half of their original values, the horizontal coordinates are stretched to twice their original values, and the graph is then shifted to the left by $\frac{\pi}{3}$ units to obtain the graph of $g(x)$, find the smallest positive period and the center of symmetry of $g(x)$;
(III) If $f(\alpha) = -1$, where $\alpha \in (\frac{\pi}{4}, \frac{\pi}{2})$, find the value of $\sin 2\alpha$.

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