
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 with the equation $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ and an eccentricity of $\frac{1}{2}$. $F\_1$ and $F\_2$ are the left and right foci of the ellipse, respectively. A line $l$ passing through $F\_2$ intersects the ellipse at points $A$ and $B$. The perimeter of $\triangle F\_1AB$ is $8$.

(I) Find the equation of the ellipse.
(II) If the slope of line $l$ is $0$, and its perpendicular bisector intersects the $y$-axis at $Q$, find the range of the $y$-coordinate of $Q$.
(III) Determine if there exists a point $M(m, 0)$ on the $x$-axis such that the $x$-axis bisects $\angle AMB$. If it exists, find the value of $m$; otherwise, explain the reason.

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