Given a natural number $n$, find the number of quadruples $(x,y,u,v)$ of integers with $1\le x,y,y,v\le n$ satisfy the following inequalities: \begin{align*} &1\le v+x-y\le n,\\ &1\le x+y-u\le n,\\ &1\le u+v-y\le n,\\ &1\le v+x-u\le n. \end{align*}