
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 $E:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\;\;(a>b>0)$ with a minor axis length of $2\sqrt{2}$ and one focus at $F_{1}(-2,0)$.<br/>$(Ⅰ)$ Find the equation and eccentricity of the ellipse $E$;<br/>$(Ⅱ)$ Let the line $l:x-my-2=0$ intersect the ellipse $E$ at points $A$ and $B$, with point $M$ lying on segment $AB$, and point $C$ being the symmetric point of $F_{1}$ with respect to point $M$. When the area of quadrilateral $AF_{1}BC$ is maximized, find the value of $m$.

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