
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 it passes through point $A(2,0)$.

(Ⅰ) Find the equation of ellipse $C$;

(Ⅱ) Suppose $M$, $N$ are two points on ellipse $C$ different from point $A$, and the product of the slopes of lines $AM$ and $AN$ equals $-\frac{1}{4}$. Determine whether line $MN$ passes through a fixed point. If so, find the coordinates of this point; if not, explain why.

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