
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 Cartesian coordinate plane $(xOy)$, there is a point $A(0,3)$ and a line $l$: $y=2x-4$. Let $C$ be a circle with radius $1$ and its center located on line $l$. If there exists a point $M$ on circle $C$ such that $|MA|=2|MO|$, find the range of the $x$-coordinate $a$ of the center $C$.

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