
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 $O$, the eccentricity is $\frac{\sqrt{3}}{2}$. 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 two distinct points $A$ and $B$, and $\overrightarrow{AP} = \lambda \overrightarrow{PB}$.
(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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