
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 P (0, 4), and Q is a moving point on the circle $x^2+y^2=8$. When Q moves on the circle, the midpoint M of PQ traces a path C. The line $l: y=kx$ intersects the path C at points A and B.  
(1) Find the equation of the trajectory C of the moving point M;  
(2) Let E (m, n) be a point on the line segment AB, and $\frac{3}{|OE|^2} = \frac{1}{|OA|^2} + \frac{1}{|OB|^2}$, express n as a function of m.

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