
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) List the first 20092009 natural numbers. Then, successively replace each number with the sum of its digits until you obtain a list of single-digit numbers. Does the list have more 4s or 5s? How many 9s are in the list?

(b) Applying the same process to the number \(3^{2009}\), that is, replacing the number with the sum of its digits, what is the resulting single-digit number?

(c) What is the resulting single-digit number for \(17^{2009}\)?

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