
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 $(xOy)$, an ellipse $(C)$ is given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with an eccentricity of $e = \sqrt{\frac{2}{3}}$. The maximum distance from any point on the ellipse to point $Q(0, 2)$ is 3.

1. Find the equation of the ellipse $(C)$.
2. Determine if there exists a point $M(m, n)$ on the ellipse 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 exists, find the coordinates of point $M$ and the corresponding area of triangle $OAB$. If not, explain the reason.

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