
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
The focal length of the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a>b>0)$ is $2\sqrt{2}$, and point $M(\sqrt{2},1)$ is a point on the ellipse $E$. A moving straight line $l$ passing through point $P(0,1)$ intersects the ellipse at points $A$ and $B$. <br/>$(1)$ Find the equation of the ellipse $E$; <br/>$(2)$ In the Cartesian coordinate system $xOy$, is there a fixed point $Q$ different from point $P$ such that $\frac{|QA|}{|QB|}=\frac{|PA|}{|PB|}$ always holds true? If yes, find the coordinates of point $Q$; if not, explain the reason.

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