Graph Algorithms · Hard problem

A one-way delivery network (a DAG) has nodes $a,b,c,d,e,f,g$ and roads $a \to b, a \to c, b \to d, c \to d, b \to e, c \to e, d \to f, e \to f, d \to g, f \to g$ (every road runs only in the direction shown). How many different routes are there from $a$ to $g$?