
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 ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$ ($a>b>0$) have its left focus at F, and its eccentricity be $\frac { \sqrt {2}}{2}$. The distance from the left intersection point with the x-axis to point F is $\sqrt {2}-1$.  
(Ⅰ) Find the equation of the ellipse;  
(Ⅱ) 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 $\frac { \sqrt {2}}{2}$, find $|AB|$.

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