AMO-Bench 2025 Problem 45

Find the largest positive integer $n$ that satisfies the following property: there exists a sequence of $n$ positive integers ${a}_{1},\cdots ,{a}_{n}$ such that ${a}_{i} \leq {2}^{2025}\left( {\forall 1 \leq i \leq n}\right)$, and for any $1 \leq i < j \leq n$, there is no non-empty proper subset $A \subset \{ i,i + 1,\cdots ,j\}$ such that $$\mathop{\sum }\limits_{{k \in A}}{a}_{k} = \mathop{\sum }\limits_{{k \in \{ i,i + 1,\cdots ,j\} - A}}{a}_{k}. \tag{1}$$