6. On the side $A C$ of triangle $A B C$, there is a point $D$ such that $A D: A C=1: 3$, and $B D+2 B C=4 A B$. The inscribed circle of the triangle with center at point $O$ intersects $B D$ at points $M$ and $N$. Find the angle $M O N$.
Answer: $2 \arccos \frac{1}{6}$.