
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 the ellipse $(C)$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 (a > b > 0)$ with a focal length of $2$ and an eccentricity of $e=\frac{1}{2}$.

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

(II) Suppose the moving line $(l)$: $y=kx+m (-\sqrt{3} < k < \sqrt{3})$ intersects the two fixed lines $(l_1)$: $\sqrt{3}x-y=0$ and $(l_2)$: $\sqrt{3}x+y=0$ at points $P$ and $Q$ respectively. If line $(l)$ always has exactly one common point with ellipse $(C)$, investigate whether the area of $\triangle OPQ$ has a minimum value. If it exists, find the minimum value; if not, explain the reason.

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