AMO-Bench 2025 Problem 4

Let $x_1,x_2,\ldots,x_{2025}$ be real numbers in the interval $[0,1]$, and let $f(x)$ be a real-valued function defined on $[0,1]$. Find the minimum possible value of $$\max_{x_1,x_2,\ldots,x_{2025}\in[0,1]} \left| \sum_{k=1}^{2025} f(x_k) - \left(\sum_{k=1}^{2025} x_k\right)^2 \right|.$$