Given that the Steiner tree problem is NP$-$hard, prove that the uncapacitated fixed$-$charge network flow problem $($FNP$)$ is also NP$-$hard. You need to reduce the Steiner tree problem to the FNP$.$ The two problems are defined as follows.

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Steiner Tree Problem: Given a graph G$=($V$,$E$),$ edge lengths, de for e in E$,$ and a subset S of vertices, find a tree in G of minimum total length that contains S$.$

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Uncapacitated Fixed$-$charge Network Flow Problem: Given a graph G$=($V$,$E$),$ each node i has demand$/$supply of bi units for i in V $($positive bi means i is a supply node, negative bi means i is a demand node and zero means none of them$).$ If an edge e is used to send flow, a fixed cost of fe is incurred. In addition, for each unit of flow there is a shipment cost ce$.$ The problem is to send flow from the supply nodes to the demand nodes with minimum total cost to satisfy all of the demand.