
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 M is on the ellipse $D: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b > 0$, the circle with center M is tangent to the x-axis at the right focus of the ellipse. If the circle M intersects the y-axis at points A and B, and $\triangle ABM$ has sides of length $\frac{2\sqrt{6}}{3}$,
1. Determine the equation of ellipse D.
2. Let P be a point on ellipse D. The line through P intersects the x-axis at F (-1, 0) and the y-axis at Q. If $\overrightarrow{QP} = 2\overrightarrow{PF}$, find the slope of line l.
3. The line through G (0, -2) intersects the left half of the ellipse N: $\frac{3x^2}{a^2} + \frac{4y^2}{b^2} = 1$ at points H and K. Another line parallel to HK passing through the right focus F1 of ellipse N intersects it at points R and S. Determine whether there exists a line GK such that |GH|•|GK|=3|RF1|•|F1S|. Explain your reasoning.

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