
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
In the plane Cartesian coordinate system $xOy$, given that the center of circle $C$ lies on the line $l: y=x-1$, and the radius is $1$.<br/>$(Ⅰ)$ If the center $C$ of the circle also lies on the line $y=-2x+8$, find the equation of circle $C$;<br/>$(Ⅱ)$ Given the point $N(0,3)$, if there exists a point $M$ on circle $C$ such that $|MN|=|MO|$, find the range of the abscissa of the center of circle $C$.

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