
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 $f(n)$. Determine the smallest integer $N'$ such that the base-eighteen representation of $g(n)$ includes the digit ‘A’ (representing 10 in decimal). Find $N'$ modulo $1000$.

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