
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 $$C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$$ has an eccentricity of $$\frac{\sqrt{3}}{2}$$, 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 ellipse C at points M and N.
(i) If the circle with chord MN as its diameter passes through the origin O, find the value of the real number m;
(ii) Suppose the point N is reflected across the x-axis to point $N_1$ (point $N_1$ does not coincide with point M), and the line $N_1M$ intersects the x-axis at point P, determine whether the area of $\triangle PMN$ has a maximum value. If it exists, find this maximum value; if not, explain why.

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