HMMT 2022 T1

Let $\left(a_{1}, a_{2}, \ldots, a_{8}\right)$ be a permutation of $(1,2, \ldots, 8)$. Find, with proof, the maximum possible number of elements of the set

$$\left\{a_{1}, a_{1}+a_{2}, \ldots, a_{1}+a_{2}+\cdots+a_{8}\right\}$$

that can be perfect squares.