
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 $f(x)=\sin x$, $\frac{\pi}{2} < \alpha < \pi$, $0 < \beta < \frac{\pi}{2}$, $f\left( \frac{\alpha}{2}-\beta\right)= \frac{2}{3}$, $f\left(\alpha- \frac{\beta}{2}+ \frac{\pi}{2}\right)=- \frac{1}{9}$.  
(1) Find the value of $\cos \frac{\alpha+\beta}{2}$.  
(2) If $g(x)= \sqrt{2}f(x+ \frac{\pi}{4})-f(2x)$, find the range of $g(x)$.

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