OMM 2021 Problem 6

Determine all non empty sets $C_1, C_2, C_3, \cdots$ such that each one of them has a finite number of elements, all their elements are positive integers, and they satisfy the following property: For any positive integers $n$ and $m$ , the number of elements in the set $C_n$ plus the number of elements in the set $C_m$ equals the sum of the elements in the set $C_{m + n}$ .

Note: We denote $\lvert C_n \lvert$ the number of elements in the set $C_n$ , and $S_k$ as the sum of the elements in the set $C_n$ so the problem's condition is that for every $n$ and $m$ : $$\lvert C_n \lvert + \lvert C_m \lvert = S_{n + m}$$ is satisfied.