
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 point $A(4,0)$ and line $l: y=2x-4$, suppose circle $C$ has a radius of $1$ and its center $C$ lies on line $l$.
$(1)$ If $CO=CA$, where $O$ is the origin, find the equation of circle $C$;
$(2)$ If the center $C$ lies on the line $y=x-1$ and a tangent line to circle $C$ passes through point $A$, find the equation of the tangent line.

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