
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 left focus of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ is $F$, the eccentricity is $\frac{\sqrt{2}}{2}$, and the distance between the left intersection of the ellipse and the $x$-axis and point $F$ is $\sqrt{2}-1$.

(1) Find the equation of the ellipse;
(2) Let $O$ be the coordinate origin, and let line $L$ pass through point $P(0,2)$, intersecting the ellipse at two distinct points $A$ and $B$. When the area of $\Delta OAB$ is $\frac{\sqrt{2}}{2}$, find $|AB|$.

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