
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 $α \in \left(0, \frac{\pi}{2}\right)$, and $\cos 2α= \frac{4}{5}$.  
$(1)$ Find the value of $\sin α + \cos α$;  
$(2)$ If $β \in \left(\frac{\pi}{2}, \pi\right)$, and $5\sin (2α+β)=\sin β$, find the magnitude of angle $β$.

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