
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 that the right focus of the ellipse (C) is F(1,0), and the point A(2,0) is on the ellipse (C). The line (l) passing through point F intersects the ellipse (C) at two distinct points M and N.
(I) Find the equation of the ellipse (C);
(II) If the slope of the line (l) is 1, find the length of the line segment MN;
(III) If the perpendicular bisector of the line segment MN intersects the y-axis at point P(0, y0), find the range of y0.

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