
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
a) How many four-digit numbers have a sum of their digits that is even?

b) A two-digit number with distinct and non-zero digits is called beautiful if the tens digit is greater than the units digit. How many beautiful numbers exist?

c) How many even four-digit numbers can we form using the digits $0,1,2,3,4,5$ without repeating any digit?

d) What is the average of all five-digit numbers that can be formed using each of the digits $1,3,5,7,$ and $8$ exactly once?

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