HMMT 2021 G3

Triangle $A B C$ has a right angle at $C$, and $D$ is the foot of the altitude from $C$ to $A B$. Points $L$, $M$, and $N$ are the midpoints of segments $A D, D C$, and $C A$, respectively. If $C L=7$ and $B M=12$, compute $B N^{2}$.