
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{3} = 1$ ($a > \sqrt{3}$), its right focus is $F$, and its right vertex is $A$. Let the eccentricity be $e$, and it satisfies $\frac{1}{|OF|} + \frac{1}{|OA|} = \frac{3e}{|AF|}$, where $O$ is the origin.
(Ⅰ) Find the equation of the ellipse $C$;
(Ⅱ) A line $l$ passing through the point $(0,1)$ intersects the ellipse at points $M$ and $N$. Find the maximum area of $\triangle OMN$.

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