A good question

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For each integer \(n\geq 4\), let \(a_n\) denote the base-\(n\) number \(0.\overline{133}_n\). The product \(a_4a_5...a_{99}\) can be expressed as \(\frac {m}{n!}\), where \(m\) and \(n\) are positive integers and \(n\) is as small as possible. What is the value of \(m\)?

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