Sequence

Algebra Level 5

The sequences \(\{a_n\}_{n=1}^\infty\) and \(\{b_n\}_{n=1}^\infty\) are defined through \(a_1=\alpha\), \(b_1=\beta\), \(a_{n+1}=\alpha a_n-\beta b_n\), and \(b_{n+1}=\beta a_n+\alpha b_n\) for all \(n\ge2\). How many pairs of real numbers \((\alpha,\beta)\) are there such that \(a_{2014}=b_1\) and \(b_{2014}=a_1\)?

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