AMC 10 2022a Problem 13

Let $\triangle ABC$ be a scalene triangle. Point $P$ lies on $\overline{BC}$ so that $\overline{AP}$ bisects $\angle BAC.$ The line through $B$ perpendicular to $\overline{AP}$ intersects the line through $A$ parallel to $\overline{BC}$ at point $D.$ Suppose $BP=2$ and $PC=3.$ What is $AD?$ $\textbf{(A) } 8 \textbf{(B) } 9 \textbf{(C) } 10 \textbf{(D) } 11 \textbf{(E) } 12$