
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

Márcio starts with a natural number and replaces it with the sum of its digits, repeating the process until he finally gets a single-digit number. For example, Márcio replaces 1784102 with 23, and then with 8. He also applies this process to lists of $N$ natural numbers, replacing each number in the list with the sum of its digits, continuing the process until he ends with a final list where each number is a single digit.

(a) Starting with $3^{2009}$, what is the final single-digit number?

(b) Starting with $17^{2009}$, what is the final single-digit number?

(c) Starting with the list of the first 20092009 natural numbers, does the final list have more 4s or 5s? How many 9s are in the final list?

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