Bob starts with an empty whiteboard. He then repeatedly chooses one of the digits $1, 2, \ldots, 9$ (uniformly at random) and appends it to the end of the currently written number. Bob stops when the number on the board is a multiple of $25$. Let $E$ be the expected number of digits that Bob writes. If $E = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $100m+n$.

Let's think step by step and output the final answer within \boxed{}.
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