Geometric Nightmare

Geometry Level 3

Diagram above shows a unit square $ABCD$ , the line $AD$ is extended to point $E$ such that $BE$ is a straight line intersecting the line extended line of $AD$ at $E$ with $F$ as a point of $BF$ that is a point on line side $CD$ such that $FE = 1$ . Let distance $DE$ be denoted as $x$ . Find the value of $x$ .

\frac{\sqrt{3}+1+\sqrt{2\sqrt{3}-1}}{3}

\frac{\sqrt{2}-2+\sqrt{2\sqrt{2}+4}}{2}

\frac{\sqrt{2}-1+\sqrt{2\sqrt{2}-1}}{2}

\frac{\sqrt{2}-1+\sqrt{2\sqrt{2}+1}}{4}