
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]


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In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $a\sin A+a\sin C\cos B+b\sin C\cos A=b\sin B+c\sin A$.  
$(1)$ Find the measure of angle $B$.  
$(2)$ If $a=2$ and $\triangle ABC$ is an acute triangle, find the range of possible values for the perimeter of $\triangle ABC$.  
$(3)$ If $b^{2}=ac$, the circumradius is $2$, the center of the circle is $O$, and $P$ is a moving point on circle $O$, find the range of possible values for $\overrightarrow{PA}⋅\overrightarrow{PB}$.

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