2024 AMC 12B Problem 23

A right pyramid has regular octagon ABCDEFGH with side length 1 as its base and apex V. Segments \overline{AV} and \overline{DV} are perpendicular. What is the square of the height of the pyramid?

\textbf{(A) }1 \qquad \textbf{(B) }\frac{1+\sqrt2}{2} \qquad \textbf{(C) }\sqrt2 \qquad \textbf{(D) }\frac32 \qquad \textbf{(E) }\frac{2+\sqrt2}{3} \qquad

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.