Question 3
As shown in Fig. 11.1, the edge length of cube $ABCD$–$EFGH$ is $2$. Take any point $P_1$ on the incircle of square $ABFE$, take any point $P_2$ on the incircle of square $BCGF$, and take any point $P_3$ on the incircle of square $EFGH$. Find the minimum and maximum of $|P_1P_2| + |P_2P_3| + |P_3P_1|$.
Fig. 11.1
Fig. 11.1