
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 that $O$ is the coordinate origin, for the function $f(x)=a\sin x+b\cos x$, the vector $\overrightarrow{OM}=(a,b)$ is called the companion vector of the function $f(x)$, and the function $f(x)$ is called the companion function of the vector $\overrightarrow{OM}$.  
$(Ⅰ)$ Let $g(x)=\sqrt{3}\sin(x-\pi)-\sin(\frac{3}{2}\pi-x)$. Find the companion vector $\overrightarrow{OM}$ of $g(x)$.  
$(Ⅱ)$ Let $\overrightarrow{ON}=(1,\sqrt{3})$ be the companion function of $f(x)$. Find the value of $\cos x$ when $f(x)=\frac{8}{5}$ and $x\in(-\frac{\pi}{3},\frac{\pi}{6})$.  
$(Ⅲ)$ From the function $g(x)$ in $(Ⅰ)$, where the ordinate remains unchanged, the abscissa is stretched to twice the original length, and then the entire graph is shifted to the right by $\frac{2\pi}{3}$ units to obtain the graph of $h(x)$. Given $A(-2,3)$ and $B(2,6)$, determine if there exists a point $P$ on the graph of $y=h(x)$ such that $\overrightarrow{AP}⊥\overrightarrow{BP}$. If it exists, find the coordinates of point $P$; if not, explain the reason.

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