
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 system $xOy$, given point $A(0,3)$ and line $l: y=2x-4$, suppose the radius of circle $C$ is $1$, and the center of circle $C$ lies on line $l$. If the moving point $M$ satisfies $|MA|=2|MO|$, and the trajectory of $M$ intersects with circle $C$. Find the range of the $x$-coordinate $a$ of the center of circle $C$.

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