
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 $f(2021) = f(31041_{\text{5}}) = 9 = 10_{\text{9}}$, and $g(2021) = \text{the digit sum of }10_{\text{9}} = 1$. Let $M$ be the least 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 $M$ is divided by $1000$.

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