
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 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 $\overrightarrow{OM}$.
$(1)$ Let $g(x)=4\cos(\frac{x}{2}+\frac{π}{3})\cdot\cos\frac{x}{2}-1$. Find the companion vector $\overrightarrow{OM}$ of $g(x)$.
$(2)$ Shift the graph of the function $g(x)$ from part (1) to the right by $\frac{π}{3}$ units, then stretch the entire graph horizontally by a factor of $2$ (vertical axis remains unchanged) 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 $\overrightarrow{AP} \perp \overrightarrow{BP}$. 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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