
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
Let the left focus of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ be $F$, and its eccentricity be $\frac{\sqrt{2}}{2}$. A line passing through point $F$ and perpendicular to the $x$-axis intersects the ellipse in a segment of length $\sqrt{2}$.

$(1)$ Find the equation of the ellipse.

$(2)$ A line passing through point $P(0, 2)$ intersects the ellipse at two distinct points $A$ and $B$. When the area of $\Delta OAB$ is maximized, find $|AB|$ and the equation of line $l$.

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