
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 $A,B$ are respectively one endpoint of the major axis and minor axis of the ellipse $C:\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1{ }(a > b > 0)$, $E,F$ are the left and right foci of the ellipse, and the intersection points of the circle with center $E$ and radius $3$ and the circle with center $F$ and radius $1$ are on the ellipse $C$, and $\left| AB \right|=\sqrt{7}$.

(I) Find the equation of ellipse $C$;

(II) Point $M$ is on the ellipse $C$, and line $ME$ is not perpendicular to the $x$-axis. Let $N$ be the other intersection point of line $ME$ and ellipse $C$, and ${M}'$ be the reflection of point $M$ about the $x$-axis. Determine whether line $N{M}'$ passes through a fixed point, and if so, find the coordinates of the fixed point; if not, explain why.

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