
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
Define a positive integer \( n \) to be a factorial tail if there is some positive integer \( m \) such that the decimal representation of \( m! \) ends with exactly \( n \) zeroes. Also, define a double factorial tail if there is some integer \( m \) where \( m! \) ends with exactly \( 2n \) zeroes. How many positive integers less than 1000 are neither factorial tails nor double factorial tails?

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