ISL 2023 A5

Let $a_{1}, a_{2}, \ldots, a_{2023}$ be positive integers such that - $a_{1}, a_{2}, \ldots, a_{2023}$ is a permutation of $1,2, \ldots, 2023$, and - $\left|a_{1}-a_{2}\right|,\left|a_{2}-a_{3}\right|, \ldots,\left|a_{2022}-a_{2023}\right|$ is a permutation of $1,2, \ldots, 2022$. Prove that $\max \left(a_{1}, a_{2023}\right) \geqslant 507$.