
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: $$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \ (a>b>0)$$ has an eccentricity of $$\frac{\sqrt{2}}{2}$$. Let F<sub>1</sub> and F<sub>2</sub> be the left and right foci of the ellipse, respectively, and let P be a point in the coordinate plane such that $$|\overrightarrow {OP}| = \frac{\sqrt{7}}{2}$$ and $$\overrightarrow {PF_{1}} \cdot \overrightarrow {PF_{2}} = \frac{3}{4}$$. Here, O is the origin of the coordinate system.  
(1) Find the equation of the ellipse C.  
(2) A line l with slope k passes through the point $$S(0, \frac{1}{3})$$ and intersects the ellipse at points A and B. Is there a fixed point M on the y-axis such that the circle with diameter AB always passes through this fixed point? If such a point exists, determine the coordinates of M; if not, please explain why.

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