
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 $\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1 (a > b > 0)$ with an eccentricity $e = \frac{\sqrt{3}}{2}$, and it passes through the point $(\frac{\sqrt{3}}{2}, 1)$.
(I) Find the equation of the ellipse;
(II) Line $l$ passes through the upper focus of the ellipse, intersecting the ellipse at points $A(x_1, y_1)$ and $B(x_2, y_2)$. Given vectors $\overrightarrow{m} = (ax_1, by_1)$ and $\overrightarrow{n} = (ax_2, by_2)$, if $\overrightarrow{m} \perp \overrightarrow{n}$, find the slope $k$ of line $l$.

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