Question 98
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 $a_{2012}$.
$$
\begin{aligned}
& a_{n+1} = a_{n} + b_{n} + \sqrt{a_{n}^{2} + b_{n}^{2}} \\
& b_{n+1} = a_{n} + b_{n} - \sqrt{a_{n}^{2} + b_{n}^{2}}
\end{aligned}
$$
Find $a_{2012}$.