
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 an ellipse $E$ with its foci on the $y$-axis and its center at the origin $O$, and its eccentricity equals $\frac{\sqrt{3}}{2}$. The perimeter of a quadrilateral, whose diagonals are the major and minor axes of the ellipse $E$, is $4\sqrt{5}$. The line $l: y=kx+m$ intersects the $y$-axis at point $P$ and intersects the ellipse $E$ at points $A$ and $B$.  
$(I)$ Find the equation of the ellipse $E$;  
$(II)$ If $\overrightarrow{AP}=3\overrightarrow{PB}$, find the range of values for $m^{2}$.

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