
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 is an ellipse (G) with the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ and an eccentricity of $e = \frac{\sqrt{2}}{2}$. The maximum distance from the point $N(0, 3)$ to any point on the ellipse is $5\sqrt{2}$.

1. Determine the equation of the ellipse (G) under these conditions.
2. Let line 'l' with a nonzero slope 'k' intersect the ellipse (G) at two distinct points, A and B. Let Q be the midpoint of AB. Can points A and B be symmetrical with respect to the line passing through points $P(0, \frac{\sqrt{3}}{3})$ and Q? If yes, determine the range of possible values for 'k'. If not, provide an explanation.

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