
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-three representation of $n$. Let $g(n)$ be the sum of the digits in the base-six representation of $f(n)$. For instance, $f(31) = f(1011_{\text{3}}) = 3$. If $f(31) = 3$ in decimal, then in base-six it's $3_6$, with digit sum $3$; hence, $g(31) = 3$. Find the smallest integer $n$ such that the base-twelve representation of $g(n)$ contains a digit or a symbol not in the set $\{0, 1, 2, ..., 9\}$. Provide $n$ modulo $500$.

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