
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 $A(0,3)$, line $l:y=2x-4$, a circle $C$ with radius $1$ and center on line $l$.<br/>$(1)$ If the center of circle $C$ also lies on the line $y=x-1$, find the equation of the tangent line passing through point $A$ to circle $C$;<br/>$(2)$ If there exists a point $M$ on circle $C$ such that $|MA|=2|MO|$, find the range of the abscissa $a$ of the center of circle $C$.

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