
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 a circle $C: x^{2}+y^{2}-2x+4y-4=0$, and a line $l$ with a slope of $1$ intersects the circle at points $A$ and $B$.
$(1)$ If line $l$ passes through the center of circle $C$, find the equation of line $l$;
$(2)$ When line $l$ is moved parallelly, find the maximum area of $\triangle CAB$ and the equation of line $l$ at this time;
$(3)$ Is there a line $l$ such that the circle with diameter $AB$ passes through the origin? If so, find the equation of line $l$; if not, explain why.

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