OLYMPIC-REVENGE 2020 Problem 4

Let $n$ be a positive integer and $A$ a set of integers such that the set $\{x = a + b\ |\ a, b \in A\}$ contains $\{1^2, 2^2, \dots, n^2\}$ . Prove that there is a positive integer $N$ such that if $n \ge N$ , then $|A| > n^{0.666}$ .