TUYMAADA 2021 S4

An $n\times n$ square ( $n$ is a positive integer) consists of $n^2$ unit squares.A $\emph{monotonous path}$ in this square is a path of length $2n$ beginning in the left lower corner of the square,ending in its right upper corner and going along the sides of unit squares. For each $k$ , $0\leq k\leq 2n-1$ , let $S_k$ be the set of all the monotonous paths such that the number of unit squares lying below the path leaves remainder $k$ upon division by $2n-1$ .Prove that all $S_k$ contain equal number of elements.