
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 a function $f(x) = 2\sin x\cos x + 2\cos\left(x + \frac{\pi}{4}\right)\cos\left(x - \frac{\pi}{4}\right)$,
1. Find the interval(s) where $f(x)$ is monotonically decreasing.
2. If $\alpha \in (0, \pi)$ and $f\left(\frac{\alpha}{2}\right) = \frac{\sqrt{2}}{2}$, find the value of $\sin \alpha$.

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