
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 $Γ:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with two vertices $M(0,2)$ and $N(0,-2)$, and its eccentricity is $\frac{\sqrt{2}}{2}$.  
$(1)$ Find the equation of the ellipse $Γ$;  
$(2)$ Let the line passing through the right focus $F$ of the ellipse $Γ$ intersect it at points $A$ and $B$, and $OA\perp OB(O$ is the origin), find the equation of the line $AB$;  
$(3)$ Let $R$ be a moving point on the ellipse $Γ$ other than $M$ and $N$, and the lines $MR$ and $NR$ intersect the line $y=-4$ at points $P$ and $Q$ respectively. Find the minimum value of $|PQ|$.

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