
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 circle $C$ has a radius of $1$ and its center $C$ lies on line $l$. If a moving point $M$ satisfies $|MA|=2|MO|$ and the trajectory of $M$ intersects with circle $C$, find the range of values for the x-coordinate $a$ of the center of circle $C$.

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