
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 $f(x) = \sin \omega x (\sin \omega x + \cos \omega x) - \frac{1}{2}$, where $\omega > 0$, and the distance between two adjacent axes of symmetry of the graph is $2\pi$.  
$(1)$ Find the intervals where $f(x)$ is monotonically increasing and the coordinates of the symmetry center of the graph of $f(x)$.  
$(2)$ Determine whether there exist acute angles $\alpha$ and $\beta$ such that $\alpha + 2\beta = \frac{2\pi}{3}$, and $f(\alpha + \frac{\pi}{2}) \cdot f(2\beta + \frac{3\pi}{2}) = \frac{\sqrt{3}}{8}$. If they exist, find the values of angles $\alpha$ and $\beta$; if not, explain the reason.

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