
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 $C$: $x^{2}+2y^{2}=8$, determine whether there exists a line $l$ with a slope of $1$ such that the chord $AB$ intercepted by $l$ on the ellipse $C$ is the diameter of a circle passing through the origin. If such a line exists, find its equation; otherwise, explain the reason.

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