
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
If \( m! \) ends with exactly \( n \) zeros, we say that \( n \) is the tail of the factorial \( m! \). Observe the examples and answer:

- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \), ends with one zero, so the tail of the factorial \( 5! \) is \( 1 \);
- \( 10! = 3628800 \), ends with two zeros, thus the tail of the factorial \( 10! \) is 2.

a) What are the tails of the factorials of \( 20! \) and \( 25! \)?

b) What is the tens digit of \( 7! + 8! + 9! + \ldots + 2018! \)?

c) Does any \( m! \) have a tail of the factorial equal to 5?

d) What is the tail of the factorial of \( 2018! \)?

e) How many positive integers less than 2018 are not tails of the factorial?

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