Let $S$ be the set of all $n$-tuples of real numbers, with the property that among the numbers $x_1,\frac{x_1+x_2}2,\ldots,\frac{x_1+x_2+\ldots+x_n}n$ the least is equal to $0$, and the greatest is equal to $1$. Determine $$\max_{(x_1,x_2,\ldots,x_n)\in S}\max_{1\le i,j\le n}(x_i-x_j)\qquad\text{and}\min_{(x_1,x_2,\ldots,x_n)\in S}\max_{1\le i,j\le n}(x_i-x_j).$$