SILK-ROAD 2015 Problem 2

Let $\left\{ {{a}_{n}} \right\}_{n \geq 1}$ and $\left\{ {{b}_{n}} \right\}_{n \geq 1}$ be two infinite arithmetic progressions, each of which the first term and the difference are mutually prime natural numbers. It is known that for any natural $n$ , at least one of the numbers $\left( a_n^2+a_{n+1}^2 \right)\left( b_n^2+b_{n+1}^2 \right)$ or $\left( a_n^2+b_n^2 \right) \left( a_{n+1}^2+b_{n+1}^2 \right)$ is an perfect square. Prove that ${{a}_{n}}={{b}_{n}}$ , for any natural $n$ .