
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 eccentricity $$\frac { \sqrt {3}}{2}$$, and point $$A(1， \frac { \sqrt {3}}{2})$$ lies on ellipse C.  
(Ⅰ) Find the equation of ellipse C;  
(Ⅱ) Suppose there exists a moving line l that intersects ellipse C at exactly one point, determine whether there exists a circle with the origin O as its center, such that this circle intersects l at two points P<sub>1</sub> and P<sub>2</sub> (both points are not on the coordinate axes), and the product of the slopes of lines OP<sub>1</sub> and OP<sub>2</sub> is a constant. If such a circle exists, find its equation; if not, explain why.

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