Let $\mathbb { R } ^ { 3 }$ denote the three-dimensional space. Take two points $P = ( 1,2,3 )$ and $Q = ( 4,2,7 )$. Let $\operatorname { dist } ( X , Y )$ denote the distance between two points $X$ and $Y$ in $\mathbb { R } ^ { 3 }$. Let
$$\begin{gathered} S = \left\{ X \in \mathbb { R } ^ { 3 } : ( \operatorname { dist } ( X , P ) ) ^ { 2 } - ( \operatorname { dist } ( X , Q ) ) ^ { 2 } = 50 \right\} \text { and } \\ T = \left\{ Y \in \mathbb { R } ^ { 3 } : ( \operatorname { dist } ( Y , Q ) ) ^ { 2 } - ( \operatorname { dist } ( Y , P ) ) ^ { 2 } = 50 \right\} \end{gathered}$$
Then which of the following statements is (are) TRUE?
(A) There is a triangle whose area is 1 and all of whose vertices are from $S$.
(B) There are two distinct points $L$ and $M$ in $T$ such that each point on the line segment $L M$ is also in $T$.
(C) There are infinitely many rectangles of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$.
(D) There is a square of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$.