
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \ (a > b > 0)$ with eccentricity $\frac{\sqrt{3}}{2}$, and its right vertex at $A(2,0)$,  
(1) Find the equation of ellipse $C$;  
(2) Does there exist a fixed point $M$ on the $x$-axis, such that any line $l$ passing through $M$ intersects the ellipse at points $B$ and $D$ and the product of slopes $k_{AB}k_{AD}$ is always $-\frac{3}{4}$? If it exists, find the coordinates of $M$; if not, provide justification.

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