BALTIC-WAY 2023 Problem 2

Let $a_{1}, a_{2}, \ldots, a_{2023}$ be positive real numbers with

$$a_{1}+a_{2}^{2}+a_{3}^{3}+\cdots+a_{2023}^{2023}=2023$$

Show that

$$a_{1}^{2023}+a_{2}^{2022}+\cdots+a_{2022}^{2}+a_{2023}>1+\frac{1}{2023} .$$