OMM 2011 Problem 3

Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system:

$$a_1^2 + a_1 - 1 = a_2$$ $$a_2^2 + a_2 - 1 = a_3$$ $$\hspace*{3.3em} \vdots$$ $$a_{n}^2 + a_n - 1 = a_1$$