
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 \( 45^6 \). 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 both \( a_1 \) divides \( a_2 \) and \( a_2 \) divides \( a_3 \) is \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m \).

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