HMMT 2023 U32

Let $A B C$ be a triangle with $\angle B A C>90^{\circ}$. Let $D$ be the foot of the perpendicular from $A$ to side $B C$. Let $M$ and $N$ be the midpoints of segments $B C$ and $B D$, respectively. Suppose that $A C=2$, $\angle B A N=\angle M A C$, and $A B \cdot B C=A M$. Compute the distance from $B$ to line $A M$.