
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 left and right vertices $M$ and $N$, and a point $P$ on the ellipse distinct from $M$ and $N$. Let the slopes of lines $PM$ and $PN$ be $k_{PM}$ and $k_{PN}$, respectively, satisfying $k_{PM} \cdot k_{PN} = -\frac{3}{4}$.

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

(II) Suppose the left focus of ellipse $(C)$ is $F(-c,0)$, and the line $AB$ passing through point $F$ intersects the ellipse at points $A$ and $B$. Let $G$ be the midpoint of $AB$, and let $D$ and $E$ be the intersection points of the perpendicular bisector of $AB$ with the $x$-axis and $y$-axis, respectively. Let $S_1$ be the area of $\triangle GFD$, $S_2$ be the area of $\triangle OED$, where $O$ is the origin. Find the range of values for $\frac{2S_1S_2}{S_1^2 + S_2^2}$.

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