Let \((a_n)_{n=1}^{\infty}\) be a sequence of \(1\)s and \(2\)s such that \(a_1 = 1\) and \(a_n\) is the number of \(2s\) between the \(n^{th}\) and \((n+1)^{th}\) \(1s\) in the sequence. So the sequence starts out like this: \[1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, ...\]

Prove (if you wish) that there exists a real number \(r\) such that \(a_n = 1\) if and only if \(n = 1 + \lfloor rk \rfloor\) for some \(k \in \mathbb{N}\).

If \(r\) can be written in simplest form as \(r = \frac{\alpha+\sqrt{\beta}}{\gamma}\) where \(\alpha, \beta, \gamma\) are coprime natural numbers, find \(\alpha^2+\beta^2+\gamma^2\).

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