
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 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 $F\_1$ about the origin. The perpendicular bisector of segment $PF\_2$ intersects with $PF\_1$ at point $M$.

(I) Find the equation of the trajectory $C$ of point $M$;

(II) Does there exist a fixed point $Q$ on the $y$-axis such that the circle with $AB$ as the diameter always passes through this point? If it exists, find the coordinates of point $Q$; if not, explain the reason. Here, the dynamic line $l$ passes through point $G(0, \frac{1}{3})$ and intersects with the trajectory $C$ of point $M$ at points $A$ and $B$.

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