A rectangular piece of paper ABCD shown below is folded so that vertices B and D coincide. Let E be the folding point on side [AB] such that $|AE| = 1$ unit.
As a result of the folding, the overlapping parts of the paper form a dark-colored equilateral triangular region DEF.
Accordingly, what is the area of the paper in square units?
A) $6\sqrt{2}$
B) $2\sqrt{2}$
C) $4\sqrt{3}$
D) $3\sqrt{3}$
E) $4\sqrt{2}$