
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \; (a > b > 0)$ passing through point $A(2,0)$, and its eccentricity is $\frac{\sqrt{3}}{2}$.

(Ⅰ) Find the equation of ellipse $C$;  
(Ⅱ) Suppose the line $y = kx + \sqrt{3}$ intersects ellipse $C$ at points $M$ and $N$. If there exists a point $P$ on the line $x = 3$ such that quadrilateral $PAMN$ is a parallelogram, find the value of $k$.

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