
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 that the left vertex of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ is $A(-2,0)$, the eccentricity is $\frac{\sqrt{3}}{2}$. The line $l$ passing through point $A$ with slope $k (k \neq 0)$ intersects the ellipse at point $D$ and intersects the $y$-axis at point $E$.  
$(Ⅰ)$ Find the equation of the ellipse.  
$(Ⅱ)$ Let point $P$ be the midpoint of $AD$.  
$(i)$ If there exists a point $Q$ on the $x$-axis such that $OP \perp EQ$ for any $k (k \neq 0)$, find the coordinates of point $Q$.  
$(ii)$ The line passing through $P$ (origin $O$) intersects the ellipse $C$ at point $M$, satisfying $\sqrt{1+4k^2} \tan ∠AMD = \frac{6}{\overrightarrow{MA} \cdot \overrightarrow{MD}}$. Find the value of the positive number $k$.

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