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

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