
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 A and B are, respectively, the endpoints of the major and the minor axes of the ellipse C: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \ (a > b > 0)$$, and E and F are the left and right foci of the ellipse. A circle centered at E with radius 3 and a circle centered at F with radius 1 intersect on ellipse C, with $$|AB| = \sqrt{7}.$$  
(1) Determine the equation of ellipse C.  
(2) If line ME is not perpendicular to the x-axis and it intersects the ellipse C at another point N, M' is the reflection of point M across the x-axis. Determine whether the line NM' passes through a fixed point, and if so, find the coordinates of that point.

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