
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 $f(x)=2\sin{x}\cos{x}+2\sqrt{3}\cos^2{x}-\sqrt{3}$.  
$(1)$ Find the extreme values of the function $f(x)$ on the interval $[0,\frac{2\pi}{3}]$.  
$(2)$ If $f(\alpha-\frac{\pi}{6})=\frac{2\sqrt{5}}{5}$, $\sin(\beta-\alpha)=\frac{\sqrt{10}}{10}$, and $\alpha\in[\frac{\pi}{4},\pi]$, $\beta\in[\pi,\frac{3\pi}{2}]$, find $\alpha +\beta$.

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