
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 a function $f(x)=a\sin x+b\cos x$, where the vector $\overrightarrow{p}=(a,b)$ is called the eigenvector of $f(x)$ and $f(x)$ is the eigenfunction of $\overrightarrow{p}$.  
$(1)$ If $g(x)=\sin(x+\frac{π}{3})+\cos(x-\frac{π}{6})$, find the eigenvector of $g(x)$.  
$(2)$ Let the eigenvectors of the vectors $\overrightarrow{p}=(\sqrt{3},-1)$ and $\overrightarrow{q}=(1,\sqrt{3})$ be $p(x)$ and $q(x)$, respectively. Define the function $h(x)=p(x)q(x)$.  
$(i)$ Find the intervals where $h(x)$ is monotonically increasing.  
$(ii)$ If the solutions of the equation $h(x)=\frac{2}{3}$ in the interval $(0,\pi)$ are $x_{1}$ and $x_{2}$, find $\cos (x_{1}-x_{2})$.

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