
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 $n$, define $f(n)$ as the sum of the digits in the base-five representation of $n$, and define $g(n)$ as the sum of the digits in the base-nine representation of $f(n)$. For example, if $f(n)$ calculates to $15$, then in base nine, it becomes $16_9$, and $g(n) = 7$. Determine the smallest number $N$ such that the base-nine representation of $g(n)$ must use at least one digit '8' or higher. What is the remainder when $N$ is divided by $1000$?

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