
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 $C_{1}$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, the distances from its vertices to the line $l_{1}$: $y = x$ are $\sqrt{2}$ and $\frac{\sqrt{2}}{2}$, respectively.

(I) Find the standard equation of $C_{1}$;

(II) Let line $l$, which is parallel to $l_{1}$, intersect $C_{1}$ at points $A$ and $B$. If a circle with $AB$ as its diameter passes through the origin, find the equation of line $l$.

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