
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
Let $S$ be the set of positive integer divisors of $30^5.$ Three numbers are chosen independently and at random with replacement from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that $a_1$ divides $a_2$, $a_2$ divides $a_3$, and $a_1, a_2, a_3$ are distinct is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m.$

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