Improved amplitude analysis of $η^\prime\toπ^+π^-π^0$ and $η^\prime\toπ^0π^0π^0$

Using a sample of $(10087\pm44)\times 10^6$ $J/ψ$ events collected with the BESIII detector at BEPCII, we perform an amplitude analysis of the decays $η^\prime\toπ^+π^-π^0$ and $η^\prime\toπ^0π^0π^0$, where we observe significant $π^\pmπ^0$ $P$-wave and $π$-$π$ $S$-wave interactions. Two different p