
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 the center of ellipse $G$ is at the origin, its major axis lies on the $x$-axis, its eccentricity is $\frac{\sqrt{3}}{2}$, and its two foci are ${{F}_{1}}$ and ${{F}_{2}}$, the sum of the distances from a point on ellipse $G$ to ${{F}_{1}}$ and ${{F}_{2}}$ is $12$. The center of circle ${{C}_{k}}$: ${{x}^{2}}+{{y}^{2}}+2kx-4y-21=0\ (k\in R)$ is point ${{A}_{k}}$.

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

(II) Find the area of $\Delta {{A}_{k}}{{F}_{1}}{{F}_{2}}$;

(III) Determine whether there exists a circle ${{C}_{k}}$ that encloses ellipse $G$, and explain the reasoning.

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