
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
Ellipse $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 \, (a > b > 0)$ passes through point $P(\sqrt{3}, 1)$ and has an eccentricity of $\dfrac{\sqrt{6}}{3}$. Let $F$ be the right focus of the ellipse, and a line passing through $F$ intersects the ellipse $C$ at points $M$ and $N$. Given a fixed point $A(-4, 0)$.

(Ⅰ) Find the equation of ellipse $C$;  
(Ⅱ) If the area of $\triangle AMN$ is $3\sqrt{3}$, find the equation of line $MN$.

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