
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 $f(x)=2\sqrt{3}\cos^2x+2\sin x \cos x-\sqrt{3}$.  
$(1)$ When $x\in[0,\frac{\pi}{2}]$, find the range of $f(x)$.  
$(2)$ If the acute angles $\alpha$ and $\beta$ satisfy $f(\frac{\alpha}{2}-\frac{\pi}{6})=\frac{8}{5}$ and $\cos(\alpha+\beta)=-\frac{12}{13}$, find $\sin \beta$.

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