
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 $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with a major axis length of $2\sqrt{2}$, $A_{1}$ and $A_{2}$ are the left and right vertices of the ellipse $C$, $P$ is a point on the ellipse different from $A_{1}$ and $A_{2}$, $O$ is the origin, and point $M$ is the midpoint of segment $PA_{2}$, and the product of the slopes of line $PA_{2}$ and $OM$ is constantly $- \frac{1}{2}$.
(Ⅰ) Find the equation of the ellipse $C$;
(Ⅱ) Suppose a line $l$ passing through point $F_{1}$ and not perpendicular to the coordinate axes intersects the ellipse at points $A$ and $B$, the perpendicular bisector of segment $AB$ intersects the $x$-axis at point $N$, and the range of the $x$-coordinate of point $N$ is $(- \frac{1}{4},0)$, find the range of the length of segment $AB$.

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