In coordinate space, for two points $\mathrm { A } ( 3 , - 3,3 ) , \mathrm { B } ( - 2,7 , - 2 )$, let $\alpha , \beta$ be the two planes that contain segment AB and are tangent to the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$. Let C and D be the points of tangency of the two planes $\alpha , \beta$ with the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ respectively. If the volume of tetrahedron ABCD is $\frac { q } { p } \sqrt { 3 }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
In coordinate space, for two points $\mathrm { A } ( 3 , - 3,3 ) , \mathrm { B } ( - 2,7 , - 2 )$, let $\alpha , \beta$ be the two planes that contain segment AB and are tangent to the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$. Let C and D be the points of tangency of the two planes $\alpha , \beta$ with the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ respectively. If the volume of tetrahedron ABCD is $\frac { q } { p } \sqrt { 3 }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]