A Prism Cut by a Plane

A regular quadrilateral prism $ABCD\text{-}A_1B_1C_1D_1$ has base $ABCD$ a square of side length $3$ and $AA_1=6$. Points $E,F,G$ lie on the segments $A_1D_1$, $AA_1$, $B_1C_1$ respectively, with $A_1F=2A_1E=2$ and $C_1G=\dfrac 32$; a point $H$ on the segment $BB_1$ satisfies $EF\parallel GH$.

1. Find the cosine of the acute dihedral angle $A_1\text{-}FH\text{-}E$. 2. Find the ratio of the volumes of the two polyhedra into which the plane $EFHG$ divides the prism.