
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, point A (0,3), the line l passes through two points (1,-2) and (3,2). Assume circle C has a radius of 1, and its center lies on line l.  
（I） Find the equation of line l;  
（II） If the chord intercepted by circle C on the x-axis is $\sqrt {3}$, find the equation of circle C;  
（III） If there exists a point M on circle C such that MA = 2MO, find the range for the x-coordinate of the center of circle C.

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