
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 eccentricity $\frac{\sqrt{3}}{2}$, the left and right foci are denoted by $F_1$ and $F_2$, respectively. A circle centered at the origin with the semi-minor axis length of the ellipse $C$ as its radius is tangent to the line $x - y + \sqrt{2} = 0$.

(I) Find the equation of the ellipse $C$.

(II) If a line $l$ that does not pass through the origin and has a defined slope intersects the ellipse $C$ at points $G$ and $H$, and the area of triangle $\triangle OGH$ is 1, the midpoint of segment $GH$ is denoted as $P$, are there two fixed points $M$, $N$ symmetric about the origin on the x-axis such that the product of the slopes of lines $PM$ and $PN$ is a constant? If they exist, find the coordinates of the two fixed points $M$, $N$ and the constant value; if they do not exist, explain why.

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