AMO-Bench 2025 Problem 19

Let $a_1 < a_2 < \cdots < a_{38}$ be positive integers with $$a_1 + a_2 + \cdots + a_{38} = 1997.$$ Define $$b_k = \min\!\left(\sum_{i=1}^{k} a_i, \sum_{i=k+1}^{38} a_i\right),$$ and let $$S = \max(b_1, b_2, \ldots, b_{38}),$$ where $b_{38} = 0$. Find the minimum possible value of $S$.