
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
Two natural numbers form a "couple" when they have the same number of digits and their sum only contains the digit 9. For example, 225 and 774 form a couple because both have three digits and \( 225 + 774 = 999 \).

a) What is the number that forms a couple with 2010?

b) How many couples are formed by two-digit numbers?

"Special couples" are couples where both numbers have the same digits, and the digits in each number are distinct. For example, 36 and 63 form a special couple, but 277 and 722 do not.

c) Give an example of a special couple with four-digit numbers.

d) Explain why there are no special couples with three-digit numbers.

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