
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 that point P (2, $\sqrt{2}$) lies on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), and the eccentricity of the ellipse is $\frac{\sqrt{2}}{2}$. Two lines $l_1$ and $l_2$ passing through point A ($-\alpha$, 0) intersect the ellipse at points E and F, and intersect the y-axis at points M and N, respectively. Points E and M are distinct, and E, F are symmetric with respect to the origin.
1. Find the equation of the ellipse.
2. Determine whether the circle with diameter MN intersects the x-axis at a fixed point Q. If so, find the coordinates of Q; otherwise, explain why.

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