
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: $$\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1 \quad (a>b>0)$$ which passes through the point $(1, \frac{2\sqrt{3}}{3})$, with its foci denoted as $F_1$ and $F_2$. The circle $x^2+y^2=2$ intersects the line $x+y+b=0$ forming a chord of length 2.
(I) Determine the standard equation of ellipse C;
(II) Let Q be a moving point on ellipse C that is not on the x-axis, with the origin O. Draw a parallel line to OQ through point $F_2$ intersecting ellipse C at two distinct points M and N.
   (1) Investigate whether $\frac{|MN|}{|OQ|^2}$ is a constant value. If so, find this constant; if not, please explain why.
   (2) Denote the area of $\triangle QF_2M$ as $S_1$ and the area of $\triangle OF_2N$ as $S_2$, 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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