
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 two foci of the ellipse $C$ are $F\_1(-2 \sqrt {2},0)$ and $F\_2(2 \sqrt {2},0)$, and the length of the major axis is $6$,
(I) Find the standard equation of the ellipse $C$;
(II) Given that the line passing through the point $(0,2)$ with a slope of $1$ intersects the ellipse $C$ at points $A$ and $B$, investigate whether the origin $O$ is on the circle with line segment $AB$ as the diameter.

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