AMO-Bench 2025 Problem 3

Find all positive integers $\mathbf{n}$ such that for any: ${a}_{n} \geq {a}_{n - 1} \geq {a}_{n - 2} \geq \cdots \cdots {a}_{2} \geq {a}_{1} > 0,$ satisfying $\mathop{\sum }\limits_{{k = 1}}^{n}{a}_{k} = \mathop{\sum }\limits_{{k = 1}}^{n}\frac{1}{{a}_{k}}$, the inequality $\mathop{\prod }\limits_{{k = 1}}^{n}{a}_{k}^{k} \geq 1$ holds.