
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 left vertex $A$ is on the circle $x^2 + y^2 = 12$.

(Ⅰ) Find the equation of ellipse $C$;

(Ⅱ) The line $l: x = my + 3 (m \neq 0)$ intersects the ellipse $C$ at points $M$ and $N$.

(i) If the circle with diameter $MN$ passes through the origin $O$, find the value of the real number $m$;

(ii) Suppose the point $N$ is symmetric to the $x$-axis at $N_1$ (point $N_1$ does not coincide with point $M$), and the line $N_1M$ intersects the $x$-axis at point $P$. Does the area of $\triangle PMN$ have a maximum value? If it exists, find this maximum value; if not, please explain why.

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