Nemotron-IMO-Bench 2026 G31

Consider a circle with center $M$ and a square $ABCD$ whose vertices lie outside the circle. Let $AA',\ldots, DD'$ be the tangents to the circle. If the segments $AA',\ldots, DD'$ are the consecutive sides of a quadrilateral $P$ in which a circle can be inscribed, proved that $P$ has an axis of symmetry.