MathNet 2023 — batch of 35 questions Question 27

Let $ABCD$ be a square, and let $M$ be the midpoint of side $BC$. Points $P$ and $Q$ lie on segment $AM$ such that $\angle BPD = \angle BQD = 135^{\circ}$. Given that $AP \lt AQ$, compute $\frac{AQ}{AP}$.