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

(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, with $Q$ being the origin. A line parallel to $OQ$ passing through $F_2$ intersects the ellipse $C$ at two distinct points $M$ and $N$.

(1) Investigate whether $\frac{|MN|}{|OQ|^2}$ is a constant. If it is, find the constant; if not, explain why.

(2) Let the area of $\triangle QF_2M$ be $S_1$ and the area of $\triangle OF_2N$ be $S_2$, and let $S = S_1 + S_2$. Find the maximum value of $S$.

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