
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{1}{2}$, a circle with the origin as its center and the short axis of the ellipse as its radius is tangent to the line $\sqrt{7}x-\sqrt{5}y+12=0$.
(1) Find the equation of the ellipse $C$;
(2) Let $A(-4,0)$, and a line $l$ passing through point $R(3,0)$ and intersecting with the ellipse $C$ at points $P$ and $Q$. Connect $AP$ and $AQ$ intersecting with the line $x=\frac{16}{3}$ at points $M$ and $N$, respectively. If the slopes of lines $MR$ and $NR$ are $k_{1}$ and $k_{2}$, respectively, determine whether $k_{1}k_{2}$ is a constant value. If it is, find this value; otherwise, explain the reason.

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