
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: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ (where $a > b > 0$) passes through the point $(\sqrt{2}, 1)$ and has an eccentricity of $\frac{\sqrt{2}}{2}$.
(I) Find the equation of ellipse C.
(II) Suppose M and N are points on ellipse C, and the product of the slopes of lines OM and ON (O being the origin) is $-\frac{1}{2}$. If a movable point P satisfies $\overrightarrow{OP} = \overrightarrow{OM} + 2\overrightarrow{ON}$, investigate whether there exist two fixed points $F_1$ and $F_2$ such that $|PF_1| + |PF_2|$ is a constant value. If they exist, determine the coordinates of $F_1$ and $F_2$. If not, explain why.

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