HMMT 2012 A6

Let $a_{0}=-2, b_{0}=1$, and for $n \geq 0$, let

$$\begin{aligned} & a_{n+1}=a_{n}+b_{n}+\sqrt{a_{n}^{2}+b_{n}^{2}} \\ & b_{n+1}=a_{n}+b_{n}-\sqrt{a_{n}^{2}+b_{n}^{2}} \end{aligned}$$

Find $a_{2012}$.