
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 $n\geq 5$, let $b_n$ denote the base-$n$ number $0.\overline{144}_n$. The product $b_5b_6 \dotsm b_{100}$ can be expressed as $\frac{p}{q!}$, where $p$ and $q$ are positive integers and $q$ is as small as possible. What is the value of $p$?

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