For each integer \( n \geq 4 \), let \( a_n \) denote the base-\( n \) number \( 0.\overline{133}_n \). The product \( a_4a_5 \dotsm a_{99} \) 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 \( m \)? Let's think step by step and output the final answer within \boxed{}.
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