
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 point O (0,0), A (0,3), and line l: y=x+1, let the radius of circle C be 1, and its center lies on line l. If there exists a point M on circle C such that $|MA|=2|MO|$, find the range of possible x-coordinates, $a$, for the center C of the circle.

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