
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: $$\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1$$ ($a > b > 0$) has an eccentricity of $$\frac {\sqrt {2}}{2}$$, and it passes through the point $$(1, \frac {\sqrt {2}}{2})$$.  
(1) Find the equation of ellipse C;  
(2) A moving line $l$ intersects ellipse C at exactly one point. Is there two fixed points on the x-axis such that the product of their distances to line $l$ equals 1? If yes, find the coordinates of these two points; if no, explain why.

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