Let $$\mathbb{R}^3$$ denote the three-dimensional space. Take two points $$P = (1, 2, 3)$$ and $$Q = (4, 2, 7)$$. Let $$dist(X, Y)$$ denote the distance between two points $$X$$ and $$Y$$ in $$\mathbb{R}^3$$. Let $$S = \{X \in \mathbb{R}^3 : (dist(X, P))^2 - (dist(X, Q))^2 = 50\}$$ and $$T = \{Y \in \mathbb{R}^3 : (dist(Y, Q))^2 - (dist(Y, P))^2 = 50\}$$. Then which of the following statements is (are) TRUE?

There is a triangle whose area is 1 and all of whose vertices are from $$S$$.

There are two distinct points $$L$$ and $$M$$ in $$T$$ such that each point on the line segment $$LM$$ is also in $$T$$.

There are infinitely many rectangles of perimeter 48, two of whose vertices are from $$S$$ and the other two vertices are from $$T$$.

There is a square of perimeter 48, two of whose vertices are from $$S$$ and the other two vertices are from $$T$$.