
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 circle C: $x^{2}+y^{2}-2x+4y-4=0$,
(1) Find the center and radius of the circle.
(2) Determine whether there exists a line $l$ with a slope of $1$ such that the chord $AB$ intercepted by circle $C$ is a diameter of a circle passing through the origin. If it exists, find the equation of line $l$; if not, explain the reason.

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