<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 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
\[ s = \sum_{i=1}^{m} a_{i} = \sum_{j=1}^{n} b_{j} \] Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
2550
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
\[ s = \sum_{i=1}^{m} a_{i} = \sum_{j=1}^{n} b_{j} \]
test