
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 that the center of an ellipse is at the origin $O$, the focus is on the $x$-axis, and the square formed by the top and bottom vertices of the ellipse and the focus has an area of $2 \sqrt {2}$.
(1) Find the equation of the ellipse;
(2) Line $l$ passes through point $P(0,2)$ and intersects the ellipse at points $A$ and $B$. When the area of triangle $AOB$ reaches its maximum value, find the equation of line $l$.

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