
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 an ellipse C: $$\frac {x^{2}}{b^{2}}+ \frac {y^{2}}{a^{2}}=1$$ (where $a>b>0$) has an eccentricity of $$\frac { \sqrt {3}}{2}$$, and the distance from one endpoint P of the minor axis of ellipse C to a focus is 2.  
(1) Find the equation of ellipse C;  
(2) Given a line l: $y=kx+ \sqrt {3}$ intersects ellipse C at points A and B, is there a real number $k$ such that the circle with diameter AB exactly passes through the origin O? If it exists, find the value of $k$; if not, explain why.

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