
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 its right focus at F(1,0) and the point A(2,0) located on the ellipse, a line l passing through the point F(1,0) intersects the ellipse C at two distinct points M and N.
(1) Find the equation of the ellipse C;
(2) Assuming that the slope of line l is 1, find the length of the line segment MN;
(3) Assuming that the perpendicular bisector of the line segment MN intersects the y-axis at point P(0, y₀), determine the range of values for y₀.

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