MathNet 2023-part-06 — 191 problems (EN–VI) Question 43

$ABCD$ is a cyclic quadrilateral with $\angle BAD = 90^\circ$ and $\angle ABC \gt 90^\circ$. $AB$ is extended to a point $E$ such that $\angle AEC = 90^\circ$. If $AB = 7$, $BE = 9$, and $EC = 12$, calculate $AD$.