
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
Call a positive integer  $x$  with non-zero digits *fruity* if it satisfies  $E(x) = 24$  where  $E(x)$  is the number of trailing zeros in the product of the digits of  $x$  defined over the positive integers. Determine the remainder when the  $30$ th smallest fruity number is divided by  $1000$ . (Trailing zeros are consecutive zeroes at the end of a number

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