
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 \( n \geq 3 \), let \( a_n \) denote the base-\( n \) number \( 0.\overline{121}_n \). The product \( a_3a_4 \dotsm a_{50} \) 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 \( m \)?

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