
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 number is called special if it does not contain the digit zero and, in addition, the sum of its digits is equal to twice the first digit. For example, the number 8161 is special because:

- None of its digits is zero;
- The sum of all its digits is $8+1+6+1=16$;
- Twice its first digit is $8 \times 2=16$.

a) Is there a special number with five digits that is even? Why? If so, provide an example.

b) What is the smallest special number with four digits?

c) What is the largest special number?

d) What is the largest special number that has all distinct digits?

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