
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 the ellipse $$C: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$$ has an eccentricity of $$\frac{\sqrt{2}}{2}$$, and the vertex of the minor axis is on the circle $x^{2} + y^{2} = 4$.

(Ⅰ) Find the equation of ellipse C;  
(Ⅱ) Given point P(-2, 3), if a line $l$ with a slope of 1 intersects ellipse C at points A and B, investigate whether an isosceles triangle ABP with AB as the base exists? If it exists, find the equation of line $l$; if not, explain why.

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