
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 an ellipse C: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ ($$a > b > 0$$) with a minor axis length of $$2\sqrt{3}$$ and an eccentricity of $$\frac{\sqrt{2}}{2}$$. Let line l be defined as x = -3.

1. Find the standard equation of ellipse C.
2. If P is the intersection point of line l and the x-axis, and D is a moving point on the ellipse above the x-axis, with E as its reflection over the x-axis, find the minimum value of $$\overrightarrow{PD} \cdot \overrightarrow{PE}$$.
3. If P is the point (-3, 3), is there a point G on ellipse C such that the quadrilateral PMGN is a parallelogram, where MN is a chord on ellipse C with a slope of $$\frac{1}{4}$$? If it exists, find the equation of the line containing chord MN. If it does not, explain why.

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