
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 $F_{1}$ 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, please explain why.

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