
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, $D$ and $E$ are the upper and right vertices of the ellipse $C$, and $S_{\triangle DEF_2} = \frac{\sqrt{3}}{2}$, eccentricity $e = \frac{1}{2}$.

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

(2) Let line $l$ pass through $F\_2$ and intersect ellipse $C$ at points $A$ and $B$. Find the minimum value of $\frac{|F\_2A| \cdot |F\_2B|}{S_{\triangle OAB}}$ (where point $O$ is the coordinate origin).

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