
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 $T$ be the set of positive integer divisors of $30^{10}$. Three numbers are chosen independently and at random with replacement from the set $T$ and labeled $d_1,d_2,$ and $d_3$ in the order they are chosen. The probability that both $d_1$ divides $d_2$ and $d_2$ divides $d_3$ is $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p$.

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