HMMT 2007 G6

Triangle $A B C$ has $\angle A=90^{\circ}$, side $B C=25, A B>A C$, and area 150. Circle $\omega$ is inscribed in $A B C$, with $M$ its point of tangency on $A C$. Line $B M$ meets $\omega$ a second time at point $L$. Find the length of segment $B L$.