
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
It is known that point P is any point on the circle $F_1$: $(x-1)^2+y^2=8$, and point $F_2$ is symmetric to point $F_1$ with respect to the origin. The perpendicular bisector of segment $PF_2$ intersects $PF_1$ and $PF_2$ at points M and N, respectively.  
(1) Find the equation of the trajectory C of point M;  
(2) A moving line $l$ passing through point $G(0, \frac{1}{3})$ intersects the trajectory C of point M at points A and B. Is there a fixed point Q on the y-axis such that the circle with diameter AB always passes through this point? If it exists, find the coordinates of point Q; if not, explain why.

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