
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 the ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$ and a minor axis length of $6$.

(I) Find the standard equation of the ellipse.
(II) Determine if there exists a line $l$ with a slope of $1$ such that $l$ intersects curve $C$ at points $A$ and $B$, and a circle with $AB$ as its diameter passes through the origin. If such a line exists, find its equation; otherwise, explain the reason.

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