2021 IMO Shortlist N3

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $\left(d_{1}, d_{2}, \ldots, d_{k}\right)$ such that for every $i=1,2, \ldots, k$, the number $d_{1}+\cdots+d_{i}$ is a perfect square.

查找具有以下属性的所有正整数 $n$：$n$ 的 $k$ 正因数具有排列 $\left(d_{1}, d_{2}, \ldots, d_{k}\right)$，使得对于每个 $i=1,2, \ldots, k$，数字 $d_{1}+\cdots+d_{i}$ 是完全平方数。