Consider the map $f: \mathbb{C} \to \mathbb{C}$ given by $w = f(z) = z^2$.
Determine the complex numbers $A, B, C,$ and $D$ that correspond to the corners of a domain in the $z$-plane whose image under the mapping $f$ is the rectangular domain in the $w$-plane bounded by the lines $u = 1, u = 10, v = 2,$ and $v = 7$.
