specify the results as follows [@Itoi97] $$\begin{aligned}
&S=\frac{1}{2}: ~~~& ~~~ P^{1/2}_{ij} = 2X + \frac{1}{2},\\
&S=1:~~~& ~~~ P^1_{ij} = X^2 + X -1,\\
&S=\frac{3}{2}:~~~& ~~~ P^{3/2}_{ij} = \frac{2}{9} X^3 + \frac{11}{18}X^2 - \frac{9}{8}X - \frac{67}{32},\\
&S = 2:~~~& ~~~ P^2_{ij} = \frac{1}{36} X^4 + \frac{1}{6} X^3 - \frac{7}{12}X^2 - \frac{5}{2} X -1.\end{aligned}$$

For studying the $S=1$ system in the current work, we introduce a site-factorized wavefunction in the time-reversal-invariant SU(3) basis, $\{ |x\ra, |y\ra, |z\ra \}$ as a complex vector ${\bf d}_j = {\bf u}_j + i {\bf v}_j$ at site $j$. Within the site-factorized wavefunction analysis, the wavefunction of $n$-sites is simply the tensor product of the wavefunction at each site. The ring exchange operator cyclically rotates the states between these sites. $$\begin{aligned}
&& P_{12\cdots n} |\Psi\ra_{12\cdots n} = P_{12\cdots n} \left[ {\bf d}_1 \otimes {\bf d}_2\otimes \cdots \otimes {\bf d}_n \right] = {\bf d}_n \otimes {\bf d}_1 \otimes {\bf d}_2 \otimes \cdots {\bf d}_{n-1}.\end{aligned}$$ Therefore, within the site-factorized wavefunction, the $n$-site ring exchange terms contribute to the Hamiltonian as $$\begin{aligned}
&& _{12\cdots n}\la \Psi| P_{12\cdots n} | \Psi \ra_{12\cdots n} = \left[ \bar{\bf d}_1 \otimes \bar{\bf d}_2 \otimes \cdots \otimes \bar{\bf d}_n \right] \cdot \left[   {\bf d}_n \otimes 