
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
For each integer $n \geq 4$, let $a_n$ denote the base-$n$ number $0.\overline{133}_n$. Calculate the product $a_4a_5 \dotsm a_{100}$ and express it as $\frac{m}{n!}$, where $m$ and $n$ are positive integers, and $n$ is as small as possible. What is the value of $m$?

Present the answer in LaTex format: \boxed{Your answer}
user