
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 $E: \frac{y^2}{a^2} + \frac{x^2}{b^2} = 1$ ($a > b > 0$) with eccentricity $\frac{\sqrt{3}}{2}$, and let $A$ and $F$ be respectively the left vertex and the upper focus point of ellipse $E$. The slope of the line $AF$ is $\sqrt{3}$. Let the line $l: y = kx + m$ intersect the y-axis at a point $P$ different from the origin and intersect ellipse $E$ at distinct points $M$ and $N$, where $\overrightarrow{MP} = \lambda \overrightarrow{PN}$.

(Ⅰ) Find the equation of ellipse $E$.
(Ⅱ) Determine if there exists a real number $m$ such that $\overrightarrow{OM} + \lambda \overrightarrow{ON} = 4 \overrightarrow{OP}$. If such an $m$ exists, find the range of $m$; if it does not, provide a justification.

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