
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 curve C where any point M on it satisfies $|MF_1|+|MF_2|=4$, where $F_1(0, -\sqrt{3})$ and $F_2(0, \sqrt{3})$,

(Ⅰ) Find the equation of curve C;  
(Ⅱ) It is known that the line $l: y=kx+\sqrt{3}$ intersects the curve C at points A and B, is there a real number k such that the circle with the line segment AB as its diameter passes through the coordinate origin O? If it exists, find the value of k; if not, please explain why.

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