
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 $E$ with one vertex at $A(0,1)$ and the focus on the $x$-axis. If the distance from the right focus of the ellipse to the line $x-y+2\sqrt{2}=0$ is $3$. <br/>$(Ⅰ)$ Find the equation of the ellipse $E$; <br/>$(Ⅱ)$ Let the line $l$ passing through point $A$ intersect the ellipse at another point $B$. Find the equation of the line $l$ when the length of chord $AB$ is maximized.

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