
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 a moving point $P(x,y)$ on a plane and two fixed points $F\_1(-\sqrt{2},0)$, $F\_2(\sqrt{2},0)$, the sum of the distances between the moving point $P$ and the fixed points $F\_1$, $F\_2$ is equal to $2\sqrt{3}$.
(I) Find the trajectory equation $C$ of the moving point $P$;
(II) Given the fixed point $E(-1,0)$, determine if there exists a value of $k$ such that the circle with $AB$ as its diameter passes through the point $E$, where $A$ and $B$ are the intersection points of the line $y=kx+2 (k \neq 0)$ and the curve $C$. If such a $k$ exists, find its value; otherwise, explain the reason.

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