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this naive super carrier added to next result could be so big over int limit, it's not practical in this question.
Not really. I did the same approach in c++ and checked how large the "super carry" could get. When calculating 100! it never reaches 100, well in the limits of an int. Anyone curious or doubting can just copy the code above and add a few lines to check.
So largest possible multiplication: 9*100
which is well under limit of int