As shown in the figure, in the quadrangular pyramid $P - A B C D$, $P A \perp$ plane $A B C D$, $A D \perp C D$, $A D \| B C$, $P A = A D = C D = 2$, $B C =$ 3. $E$ is the midpoint of $P D$, and point $F$ is on $P C$ such that $\frac { P F } { P C } = \frac { 1 } { 3 }$. (I) Prove that: $C D \perp$ plane $P A D$; (II) Find the cosine of the dihedral angle $F - A E - P$; (III) Let point $G$ be on $P B$ such that $\frac { P G } { P B } = \frac { 2 } { 3 }$. Determine whether line $A G$ lies in plane $A E F$, and explain the reason.
2
As shown in the figure, in the quadrangular pyramid $P - A B C D$, $P A \perp$ plane $A B C D$,
$A D \perp C D$, $A D \| B C$, $P A = A D = C D = 2$, $B C =$ 3. $E$ is the midpoint of $P D$, and point $F$ is on $P C$ such that $\frac { P F } { P C } = \frac { 1 } { 3 }$.
(I) Prove that: $C D \perp$ plane $P A D$;
(II) Find the cosine of the dihedral angle $F - A E - P$;
(III) Let point $G$ be on $P B$ such that $\frac { P G } { P B } = \frac { 2 } { 3 }$. Determine whether line $A G$ lies in plane $A E F$, and explain the reason.