A Triangle on Three Incircles of a Cube

As shown in the figure, the cube $ABCD\text{-}EFGH$ has edge length $2$, where $E$, $F$, $G$, $H$ lie directly above $A$, $B$, $C$, $D$ respectively. A point $P_1$ is chosen arbitrarily on the incircle of the square face $ABFE$, a point $P_2$ on the incircle of the square face $BCGF$, and a point $P_3$ on the incircle of the square face $EFGH$.

Find the minimum and maximum values of