
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 a circle C: $x^2+y^2+2x-4y+3=0$.
- (I) Find the coordinates of the center C and the radius r;
- (II) Given a line l that does not pass through the origin and is tangent to circle C, and the intercepts on the x-axis and y-axis are equal, find the equation of line l;
- (III) From a point P (x, y) outside circle C, draw a tangent line to the circle at point M, with O being the origin, and $|MP|=|OP|$, find the equation of the trajectory of point P.

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