
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 an ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a>b>0)$ with its left focus as F, lower vertex as A, and upper vertex as B. $\triangle FAB$ is an equilateral triangle.
(I) Find the eccentricity of the ellipse;
(II) Let line l: x=-a pass through point A and intersect with the ellipse at point C (C is different from A). The perpendicular bisector of segment AC intersects with line l at point P and with line AC at point Q. If $|PQ|= \frac{7}{4}|AC|$.
(i) Find the value of k (the slope of line l);
(ii) Given point $M(- \frac{4}{5},- \frac{4}{5})$, and point N is on the ellipse. If quadrilateral AMCN is a parallelogram, find the equation of the ellipse.

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