
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
Let $f(x)=\sin(2x-\frac{π}{6})+\sin(2x+\frac{π}{3})$.  
$(1)$ When $x\in [0,\frac{5π}{8}]$, find the range of $f(x)$.  
$(2)$ If $α∈(\frac{π}{6},π)$ and $f(\frac{α}{2})=\frac{1}{2}$, find the value of $\sin(2α+\frac{π}{6})$.

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