Seven small towns—H, I, J, K, L, M, N—are serviced by three roads—Routes 1, 2, and 3.
Route 1 ends at N and L, and passes through M only.
Route 2 starts at H. Then passes through L and I, and ends back at H.
Route 3 ends at N and K, and passes through J only.
Two towns are directly connected if a person can drive from one of the towns to the other without passing through any other town.
If a new road were built directly connecting H and K, then the maximum number of paths connecting L to J, which do not pass through any town more than once, would be