
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 vectors $\overrightarrow{a}=({\sqrt{3},1})$ and $\overrightarrow{b}=({cosx,sinx})$, where $x\in \left(0,\pi \right)$.<br/>$(1)$ If $\overrightarrow{a}⊥\overrightarrow{b}$, find the value of $x$;<br/>$(2)$ If $f(x)=\overrightarrow{a}•\overrightarrow{b}$, and $f(α)=\frac{{2\sqrt{2}}}{3}$, find the value of $sin({2α+\frac{π}{6}})$.

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