Question 134
Let $ABC$ be a triangle with $\angle BAC \gt 90^{\circ}$. Let $D$ be the foot of the perpendicular from $A$ to side $BC$. Let $M$ and $N$ be the midpoints of segments $BC$ and $BD$, respectively. Suppose that $AC = 2$, $\angle BAN = \angle MAC$, and $AB \cdot BC = AM$. Compute the distance from $B$ to line $AM$.