HMMT 2014 A9

Given that $a, b$, and $c$ are complex numbers satisfying

$$\begin{aligned} a^{2}+a b+b^{2} & =1+i \\ b^{2}+b c+c^{2} & =-2 \\ c^{2}+c a+a^{2} & =1, \end{aligned}$$

compute $(a b+b c+c a)^{2}$. (Here, $i=\sqrt{-1}$.)