A factory produces a specific type of widget in batches. Each batch contains a number of widgets that follows a specific pattern: the first batch contains 1 widget, the second batch contains 11 widgets, the third batch contains 111 widgets, the fourth batch contains 1111 widgets, and so on. If the factory continues this pattern and produces a total of 8 batches, what is the sum of the widgets produced in all 8 batches? Express your answer in the simplest form. Let's think step by step and output the final answer within \boxed{}.
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