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

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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