AMC 12 2024b 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?

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