
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
In the Cartesian coordinate system xOy, it is known that the ellipse C has the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a>b>0)$ with eccentricity $e=\frac{\sqrt{a^2 - b^2}}{a}=\frac{\sqrt{3}}{2}$, and the maximum distance from any point on ellipse C to point Q(0,2) is 3.

(1) Determine the equation of ellipse C;
(2) On the ellipse C, is there a point M(m, n) such that the line $l: mx + ny = 1$ intersects the circle $O: x^2 + y^2 = 1$ at two distinct points A and B, and the area of triangle △OAB is maximized? If such a point M exists, find the coordinates of M and the area of △OAB; if not, explain why.

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