
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 $f(x)=\sin (2x-\frac{\pi }{4})$.

1. Find the interval(s) where $f(x)$ is monotonically decreasing.
2. If $\alpha \in (\frac{\pi }{2},\pi )$ and $f(\frac{\alpha }{2}+\frac{\pi }{4})=\frac{2}{3}\cos (\alpha +\frac{\pi }{4})\cos 2\alpha $, find the value of $\sin \alpha -\cos \alpha $.

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