
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$, the left and right foci are $F_1$ and $F_2$, respectively. A circle with the origin $O$ as its center and the ellipse's minor axis as its radius is tangent to the line $x - y + 2 = 0$.

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

(II) Let $Q$ be a moving point on the ellipse $C$ not on the $x$-axis. A line parallel to $OQ$ passing through $F_2$ intersects the ellipse $C$ at two distinct points $M$ and $N$. Let $S_1$ denote the area of $\triangle QF_2M$ and $S_2$ denote the area of $\triangle OF_2N$. Find the maximum value of $S_1 + S_2$.

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