
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 $\overrightarrow{a}=(\sin x,\cos x)$, $\overrightarrow{b}=(\cos x,\sqrt{3}\cos x)$, and a function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}-\frac{\sqrt{3}}{2}$.

$(1)$ If $f\left(\frac{x_{0}}{2}\right)=-\frac{1}{3}$, and $x_{0}\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$, find the value of $\sin x_{0}$.

$(2)$ Given $A(-3,2)$, $B(3,10)$, shifting the graph of $f(x)$ to the left by $\frac{\pi}{12}$ units results in the graph of a function $g(x)$. Does there exist a point $P$ on the graph of $g(x)$ such that $AP\perp BP$? If so, find the coordinates of all points $P$ that satisfy this condition; if not, explain the reason.

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