
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), its right focus is F(2, 0). A line passing through point F intersects the ellipse at points M and N, and the midpoint of MN is ($$1, \frac {\sqrt {2}}{2}$$).

1. Find the equation of C.
2. Suppose a line does not pass through point P(0, b) and intersects C at points A and B. If the sum of the slopes of lines PA and PB is l, determine whether the line passes through a fixed point. If it does, find the fixed point. If not, provide a reason.

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