How well do you know \(\sqrt{2017}?\)

Algebra Level 3

\[f(x) = \begin{cases} 0 + \beta \ : \ \lfloor x \rfloor \equiv 0 \mod 3 \\ 1 + \beta \ : \ \lfloor x \rfloor \equiv 1 \mod 3 \\ 2 + \beta \ : \ \lfloor x \rfloor \equiv 2 \mod 3 \\ \end{cases}\]

There is a unique value of \(\beta\) such that \[\displaystyle \sum_{n=1}^{\infty} \frac{f \left( 3^{n}\sqrt{2017} \right)}{3^{n}} = 0.\] This value of \(\beta \) can be expressed as \( a-b\sqrt{c},\) where \(a,\ b,\) and \(c\) are positive integers and \(c\) is square-free. Find \( a + b + c .\)

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