HMMT 2024 T1

Let $a_{1}, a_{2}, a_{3}, \ldots, a_{100}$ be integers such that

$$\frac{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+\cdots+a_{100}^{2}}{a_{1}+a_{2}+a_{3}+\cdots+a_{100}}=100 .$$

Determine, with proof, the maximum possible value of $a_{1}$.