
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with its left and right foci as $F_1$ and $F_2$, respectively, and its left vertex as $A$. The eccentricity is $\dfrac{\sqrt{2}}{2}$, and point $B$ is a moving point on the ellipse. The maximum area of $\triangle ABF_1$ is $\dfrac{\sqrt{2}-1}{2}$.

(Ⅰ) Find the equation of ellipse $C$;

(Ⅱ) Suppose a line $l$ passing through point $F_1$ intersects the ellipse $C$ at two distinct points $M$ and $N$, and the perpendicular bisector of segment $MN$ is $l′$. If line $l′$ intersects line $l$ at point $P$ and intersects the line $x=2$ at point $Q$, find the minimum value of $\dfrac{|PQ|}{|MN|}$.

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