
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 an ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with foci $F_{1}$ and $F_{2}$, and vertex $B$ at the bottom, passing through $P(\frac{8}{3},\frac{-b}{3})$, and $BF_{2}\perp PF_{2}$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ Let the line passing through $M(0,1)$ intersect the ellipse $C$ at points $E$ and $D$, and $Q$ be a moving point on the coordinate plane. The reciprocals of the slopes of lines $QE$, $QM$, and $QD$ form an arithmetic sequence. Investigate whether point $Q$ lies on a certain fixed line. If it exists, find the equation of that fixed line. If it does not exist, explain the reason.

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