
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: $$\frac{x^2}{a^2} + y^2 = 1 \ (a > 0)$$, where F<sub>1</sub> and F<sub>2</sub> are its left and right foci respectively. The circle with diameter F<sub>1</sub>F<sub>2</sub> intersects the ellipse C at exactly two points.
1. Find the equation of ellipse C.
2. Suppose a line l, not perpendicular to the coordinate axes, passes through point F<sub>1</sub> and intersects the ellipse at points A and B. The perpendicular bisector of line segment AB intersects the x-axis at point P. If the x-coordinate of point P is in the range $(-\frac{1}{4}, 0)$, find the range of the length of segment AB.

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