How to Find a Period-Three Cycle

Question

I have learnt that a period-two cycle, say of the logistic map $$x_{k+1}=rx_k-rx_k^2, \ \ x_k\in (0,1), \ \ r\in (0,4),$$ can be found by finding $x_k=x_{k+2}$ where $x_k\neq x_{k+1}$. For this logistic map, the period-two cycle is $$x_0=\frac{1+r+\sqrt{(r-3)(r+1)}}{2r} \\ x_1=\frac{1+r-\sqrt{(r-3)(r+1)}}{2r},$$ where $r\geq 3$. I am curious how the three cycle is found. Do we simply search for $x_k=x_{k+3}$, where $x_k\neq x_{k+1}$ and $x_k\neq x_{k+2}$? I could not find an example in my lecture notes, and I am wondering how a period-three cycle could be obtained in general (assuming this can be done be hand).