As shown in the figure for question (20), in the triangular pyramid $\mathrm { P } - \mathrm { ABC }$, plane $\mathrm { PAC } \perp$ plane $\mathrm { ABC }$, $\angle \mathrm { ABC } = \frac { \pi } { 2 }$. Points $D$ and $E$ lie on segment $AC$ with $\mathrm { AD } = \mathrm { DE } = \mathrm { EC } = 2$, $\mathrm { PD } = \mathrm { PC } = 4$. Point $F$ lies on segment $AB$ with $\mathrm { EF } \parallel \mathrm { BC }$ . (I) Prove that $\mathrm { AB } \perp$ plane $PFE$ . (II) If the volume of the quadrangular pyramid $\mathrm { P } - \mathrm { DFBC }$ is 7, find the length of segment $BC$.
As shown in the figure for question (20), in the triangular pyramid $\mathrm { P } - \mathrm { ABC }$, plane $\mathrm { PAC } \perp$ plane $\mathrm { ABC }$, $\angle \mathrm { ABC } = \frac { \pi } { 2 }$. Points $D$ and $E$ lie on segment $AC$ with $\mathrm { AD } = \mathrm { DE } = \mathrm { EC } = 2$, $\mathrm { PD } = \mathrm { PC } = 4$. Point $F$ lies on segment $AB$ with $\mathrm { EF } \parallel \mathrm { BC }$ .
(I) Prove that $\mathrm { AB } \perp$ plane $PFE$ .
(II) If the volume of the quadrangular pyramid $\mathrm { P } - \mathrm { DFBC }$ is 7, find the length of segment $BC$.