11th European Girls' Mathematical Olympiad | EGMO 2022

Problem 3. An infinite sequence of positive integers $a_1, a_2, \dots$ is called $good$ if
(1) $a_1$ is a perfect square, and
(2) for any integer $n \ge 2$ , $a_n$ is the smallest positive integer such that $na_1 + (n-1)a_2 + \dots + 2a_{n-1} + a_n$ is a perfect square.
Prove that for any good sequence $a_1, a_2, \dots$ , there exists a positive integer $k$ such that $a_n=a_k$ for all integers $n \ge k$ .