Graph Algorithms · Superchallenge problem

Seven tasks $1$–$7$ have prerequisites $1\to2, 1\to3, 1\to4, 2\to5, 3\to5, 3\to6, 4\to6, 5\to7, 6\to7$ (the tail must be done before the head, and this forces further relations transitively). A CHAIN is a set of tasks that are pairwise forced into a specific order by these prerequisites. What is the fewest number of chains needed so that every one of the $7$ tasks belongs to exactly one chain?