Core Idea - Max Flow Equals Path Count

Each unit of flow corresponds to one edge-disjoint path. If max flow is $k$, you can find $k$ paths from $s$ to $t$ that do not share edges. This is a classical result. Compute max flow with unit capacities. The flow value is the number of disjoint paths.

This answers the first part of the problem. To extract the paths (the second part), you need to decompose the flow into individual paths. Flow decomposition converts the abstract flow into concrete routes.