
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 that $F_1$ and $F_2$ are the left and right foci of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, respectively, and $D$ and $E$ are the top and right vertices of the ellipse $C$, respectively, with the area of $\triangle DEF_2 = \frac{\sqrt{3}}{2}$, and the eccentricity $e = \frac{1}{2}$.

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

  (II) Suppose a line $l$ passing through $F_2$ intersects the ellipse $C$ at points $A$ and $B$, find the minimum value of $\frac{|F_2A| \cdot |F_2B|}{S_{\triangle OAB}}$.

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