
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 line $y=-x+1$ intersects with the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ at points $A$ and $B$, and the midpoint of segment $AB$ lies on the line $l: x-2y=0$.

(Ⅰ) Find the eccentricity of this ellipse;  
(Ⅱ) If the symmetric point of the right focus of the ellipse with respect to line $l$ lies on the circle $x^2+y^2=4$, find the equation of this ellipse.

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