
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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
\[ s = \sum_{i=1}^{m} a_{i} = \sum_{j=1}^{n} b_{j} \]

Present the answer in LaTex format: \boxed{Your answer}
user