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

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

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