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Suppose $w(t)=4 e^{i t}, x(t)=3 e^{i(t+\pi / 3)}, y(t)=3 e^{i(t-\pi / 3)}$ and $z(t)=3 e^{i(t+\pi)}$ are points that move in the complex plane as the time $t$ varies in $(-\infty, \infty)$. Let $c(t)$ be the point in the complex plane such that $|w(t)-c(t)|^2+|x(t)-c(t)|^2+|y(t)-c(t)|^2+|z(t)-c(t)|^2$ is minimum. For each value of $t$, the point $c(t)$ is unique, but $c(t)$ moves at constant speed as $t$ varies. At what speed? That is, what is $\left|\frac{\mathrm{d}}{\mathrm{d} t} c(t)\right|?$

1. $\frac{1}{2 \pi}$
2. $2 \pi$
3. $\sqrt{3} \pi$
4. $\frac{1}{\sqrt{3} \pi}$
5. $1$