Bulgarian Spring Tournament

Points $A$, $B$, $Y$ and $C$ lie in this order on circle $k$ with center $O$, such that $BC=2$ cm, $\angle BAY=42^{\circ }$ and $\angle CAY=78^{\circ }$. It is known that the circle $\omega$ through the points $A$, $O$ and $B$ is tangent to the line $BY$.

The circle through the points $A$ and $C$, tangent to the line $CY$, intersects $\omega$ for second time at the point $N$. To be found:

a) the length of the segment $BO$; b) the size of the angle $\angle YAN$.