AMC 12 2013b Problem 17

Let $a,b,$ and $c$ be real numbers such that

$$a+b+c=2, \text{ and}$$ $$a^2+b^2+c^2=12$$

What is the difference between the maximum and minimum possible values of $c$?

$\textbf{(A) }2 \textbf{ (B) }\frac{10}{3} \textbf{ (C) }4 \textbf{ (D) }\frac{16}{3} \textbf{ (E) }\frac{20}{3}$