Middle European Mathematical Olympiad 2021 Problem T-3

Let $n$, $b$ and $c$ be positive integers. A group of $n$ pirates wants to fairly split their treasure. The treasure consists of $c \cdot n$ identical coins distributed over $b \cdot n$ bags, of which at least $n - 1$ bags are initially empty. Captain Jack inspects the contents of each bag and then performs a sequence of moves. In one move, he can take any number of coins from a single bag and put them into one empty bag. Prove that no matter how the coins are initially distributed, Jack can perform at most $n - 1$ moves and then split the bags among the pirates such that each pirate gets $b$ bags and $c$ coins.