
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
For each positive integer \( n \), let \( f(n) \) be the sum of the digits in the base-five representation of \( n \) and let \( g(n) \) be the sum of the digits in the base-nine representation of \( f(n) \). For example, if \( f(2021) = f(31041_5) = 9 = 10_9 \), then \( g(2021) = \text{the digit sum of } 10_9 = 1 \). Determine the smallest value of \( n \) such that the base-seventeen representation of \( g(n) \) cannot be expressed using only the digits \( 0 \) through \( 9 \). Find the remainder when \( N \) is divided by \( 1000 \).

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