
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 circle C: $(x^2+y^2-4x+2y+m=0)$ intersects with the y-axis at points A and B, forming a right angle ∠ACB (C is the center of the circle). A line passing through point P(0,2) with a slope of k intersects circle C at points M and N.

(I) Find the value of the real number m.
(II) If $|MN| \geqslant 4$, find the range of k.
(III) If the vector $\overrightarrow{OM} + \overrightarrow{ON}$ is collinear with the vector $\overrightarrow{OC}$ (O is the origin), find the value of k.

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