
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 $\overrightarrow{a} = (\sqrt{2}\cos\omega x, 1)$ and $\overrightarrow{b} = (2\sin(\omega x + \frac{\pi}{4}), -1)$ (where $\frac{1}{4} \leq \omega \leq \frac{3}{2}$), the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$, and the graph of $f(x)$ has an axis of symmetry at $x = \frac{5\pi}{8}$.
1. Find the value of $f(\frac{3}{4}\pi)$;
2. If $f(\frac{\alpha}{2} - \frac{\pi}{8}) = \frac{\sqrt{2}}{3}$, $f(\frac{\beta}{2} - \frac{\pi}{8}) = \frac{2\sqrt{2}}{3}$, and $\alpha, \beta \in (-\frac{\pi}{2}, \frac{\pi}{2})$, find the value of $\cos(\alpha - \beta)$.

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