
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 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$) with its left focus at F and the eccentricity $e = \frac{\sqrt{2}}{2}$, the line segment cut by the ellipse from the line passing through F and perpendicular to the x-axis has length $\sqrt{2}$.
(Ⅰ) Find the equation of the ellipse.
(Ⅱ) A line $l$ passing through the point P(0,2) intersects the ellipse at two distinct points A and B. Find the length of segment AB when the area of triangle OAB is at its maximum.

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