
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 $\frac{1}{4}\leqslant \omega\leqslant \frac{3}{2}$, and 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 $f( \frac{\alpha}{2}- \frac{\pi}{8})= \frac{\sqrt{2}}{3}$ and $f( \frac{\beta}{2}- \frac{\pi}{8})= \frac{2\sqrt{2}}{3}$, 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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