
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 $(E)$ that passes through point $A(2,3)$, with its axes of symmetry being the coordinate axes, foci $F_{1}$, $F_{2}$ on the $x$-axis, an eccentricity $e= \frac {1}{2}$, and the bisector of $\angle F_{1}AF_{2}$ being on line $l$.
(I) Find the equation of ellipse $(E)$;
(II) Let $Q$ be the intersection point of line $l$ and the $x$-axis. Find the coordinates of point $Q$ and the equation of line $l$;
(III) Determine whether there exist two distinct points on ellipse $(E)$ that are symmetric about line $l$. If they exist, find them; if not, explain the reason.

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