
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 the ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $(1, \dfrac{2\sqrt{3}}{3})$, with the left and right foci being $F_1$ and $F_2$ respectively. The circle $x^2 + y^2 = 2$ intersects with the line $x + y + b = 0$ at a chord of length $2$.

(Ⅰ) Find the standard equation of the ellipse $C$;

(Ⅱ) Let $Q$ be a moving point on the ellipse $C$ that is not on the $x$-axis, $O$ be the origin. A line parallel to $OQ$ passing through $F_2$ intersects the ellipse $C$ at two distinct points $M$ and $N$. Investigate whether the value of $\dfrac{|MN|}{|OQ|^2}$ is a constant. If it is, find this constant; if not, please explain why.

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