
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 \(36^7\). 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}
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