HMMT 2022 C8

Random sequences $a_{1}, a_{2}, \ldots$ and $b_{1}, b_{2}, \ldots$ are chosen so that every element in each sequence is chosen independently and uniformly from the set $\{0,1,2,3, \ldots, 100\}$. Compute the expected value of the smallest nonnegative integer $s$ such that there exist positive integers $m$ and $n$ with

$$s=\sum_{i=1}^{m} a_{i}=\sum_{j=1}^{n} b_{j} .$$