
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
An ellipse $Γ: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with a focal length of $2c$ satisfies the condition that $a$, $b$, and $c$ form an arithmetic progression. Such an ellipse $Γ$ is called an "arithmetic ellipse."  
$(1)$ Find the eccentricity of $Γ$;  
$(2)$ A line $l$ passing through $D(0, -a)$ intersects $Γ$ at exactly one point. Find the value of the slope $k$ of this line;  
$(3)$ Let point $A$ be the right vertex of the ellipse, $P$ be any point on the ellipse other than $A$, and $Q$ be the symmetric point of $P$ with respect to the origin $O$ (also different from $A$). Lines $AP$ and $AQ$ intersect the $y$-axis at points $M$ and $N$, respectively. Determine whether the circle with diameter $MN$ passes through a fixed point. Explain the reason.

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