
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
In the Cartesian coordinate system $xOy$, it is known that the ellipse $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $(2, \sqrt{2})$, and its eccentricity is $\frac{\sqrt{2}}{2}$.

(Ⅰ) Find the equation of $E$;  
(Ⅱ) A line $l$ passing through the left focus $F$ of $E$ with a non-zero slope intersects $E$ at points $A$ and $B$. The midpoint of the line segment $AB$ is $C$, and the line $OC$ intersects the line $x=-4$ at point $D$. If $\triangle ADF$ is an isosceles right triangle, find the equation of $l$.

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