
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
Given a positive integer \( t \), determine the number of integers \( b > 0 \) for which \( 30t \) is divisible by \( b! \). (If \( n \) is a positive integer, the symbol \( n! \) (read " \( n \) factorial") represents the product of the integers from 1 to \( n \). For example, \( 4! = (1)(2)(3)(4) \) or \( 4! = 24 \).)

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