Directed Graphs and Network Flows · Superchallenge problem

A network mixes one-way and two-way links: $s\to a$, $s\to e$, $s\leftrightarrow c$ (two-way), $a \leftrightarrow b$ (two-way), $e \to b$, $b \to t$, $c \leftrightarrow t$ (two-way), each usable at most once in a chosen direction. What is the maximum number of pairwise edge-disjoint paths from $s$ to $t$?