Like Base System?

Number Theory Level pending

For each integer \(n\geq 4 \), let \(a_n\) denote the base-\(n\) number \( (0.133133133\ldots)_n\). The product \(a_4a_5\cdots a_{99} \) can be expressed as \(\dfrac{m}{n!} \), where \(m\) and \(n\) are positive integers and \(n\) is as small as possible. What is the value of \(m\)?

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