Nemotron-IMO-Bench 2026 A38

Let $f: [0,1]\to \mathbf{R}$ be a non-decreasing function. What is the maximal possible value of the expression $$\sum_{k=1}^n f(|x_k-\frac{2k-1}{2n}|)$$ over all possible $n$-tuples $(x_1,\ldots, x_n)$ such that $0\leq x_1\leq x_2\leq\ldots\leq x_n\leq 1$?