ISL 2021 C4

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a path from $X$ to $Y$ is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called diverse if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{A B}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{B A}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{A B}=N_{B A}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$.