
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 $f(x)=2\sin (x+\frac{π}{4})\cos (x+\frac{π}{4})+2\sqrt{3}\sin x\cos x$. Find:<br/>$(1)$ The interval where $f(x)$ is monotonically increasing;<br/>$(2)$ If $f({\frac{α}{2}})=\frac{8}{5}$, and $α∈({\frac{π}{2},π})$, find the value of $\sin \alpha$.

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