
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 the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\frac{\sqrt{3}}{2}$, and the area of the rhombus formed by its four vertices is $4$. Circle $M: (x+1)^2 + y^2 = r^2 (0 < r < 1)$. A tangent line to circle $M$ passing through the top vertex $A$ of ellipse $C$ intersects the ellipse $C$ at points $B$ and $D$ (different from point $A$), and the slopes of lines $AB$ and $AD$ are $k_1$ and $k_2$, respectively.  
$(1)$ Find the equation of ellipse $C$;  
$(2)$ As $r$ varies, $(i)$ find the value of $k_1 \cdot k_2$; $(ii)$ determine whether line $BD$ passes through a fixed point. If yes, find the fixed point; if no, explain why.

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