Nemotron-IMO-Bench 2026 N47

Prove that every positive integer $n$, except a finite number of them, can be represented as a sum of $2004$ positive integers: $n=a_1+a_2+\ldots +a_{2004}$, where $1\le a_1<a_2<\ldots <a_{2004}$, and $a_i \mid a_{i+1}$ for all $1\le i\le 2003$.