Graph Algorithms · Superchallenge problem

A weighted graph has vertices $s,a,b,t$ and edges $sa{:}3, sb{:}10, at{:}9, bt{:}4$, together with an edge $ab$ whose weight $w$ is a positive integer. For how many positive integer values of $w$ is $s,a,b,t$ the UNIQUE shortest route from $s$ to $t$?