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

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