
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 $(C)$ with its center at the origin, a short axis length of $4$, and one focus coinciding with a focus of the parabola $y^{2}=4 \sqrt {5}x$.
(1) Find the equation of the ellipse $(C)$.
(2) Given a line $(l)$ passing through the fixed point $M(2,0)$ with a slope not equal to $0$, and intersecting the ellipse $(C)$ at points $A$ and $B$. Determine if there exists another fixed point $P$ on the $x$-axis such that $PM$ always bisects $\angle{APB}$. If such a point exists, find its coordinates; otherwise, explain the reason.

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