
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 the equation of circle C: x^2 + (y-4)^2 = 4, and point O is the origin. Line l: y=kx intersects circle C at points M and N.
(I) Find the range of k;
(II) Find the trajectory of the midpoint P of MN;
(III) Let Q(m, n) be a point on segment MN, and $\frac {2}{|OQ|^{2}}= \frac {1}{|OM|^{2}}+ \frac {1}{|ON|^{2}}$. Express n as a function of m.

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