CN-TST 2023 Problem 17

Whether there are integers $a_1$ , $a_2$ , $\cdots$ , that are different from each other, satisfying: (1) For $\forall k\in\mathbb N_+$ , $a_{k^2}>0$ and $a_{k^2+k}<0$ ; (2) For $\forall n\in\mathbb N_+$ , $\left| a_{n+1}-a_n\right|\leqslant 2023\sqrt n$ ?