GULF 2014 Problem 1

A sequence $a_0,a_1,a_2,\cdots$ satisfies the conditions $a_0 = 0$ , $a_{n-1}^2 - a_{n-1} = a_n^2 + a_n$ 1) determine the two possible values of $a_1$ . then determine all possible values of $a_2$ . 2)for each $n$ , prove that $a_{n+1}=a_n+1$ or $a_{n+1} = -a_n$ 3)Describe the possible values of $a_{1435}$ 4)Prove that the values that you got in (3) are correct