
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with its right focus $F$, eccentricity $\frac{\sqrt{2}}{2}$, and a line segment of length $\sqrt{2}$ intercepted by the ellipse on the line passing through point $F$ and perpendicular to the $x$-axis. $O$ is the coordinate origin.
(I) Find the equation of ellipse $C$;
(II) Let line $AB$ passing through point $M(0,2)$ intersect ellipse $C$ at points $A$ and $B$. Find the maximum area of $\triangle AOB$.

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