KöMaL 2025 C1854

Circles $k_1$ and $k_2$ intersect each other in points $A$ and $B$ such that circle $k_2$ passes through center $K$ of circle $k_1$. Let's choose point $S$ on circle $k_2$ such that $S$ is closer to point $A$ than to point $B$, and it's not in the interior of circle $k_1$. Let line $BS$ intersect circle $k_1$ at point $D$ for the second time, and let line $AD$ intersect circle $k_2$ at point $T$ for the second time. Prove that $KT$ is perpendicular to $BS$.