
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 $h(n)$ be the sum of the digits in the base-six representation of $n$ and let $j(n)$ be the sum of the digits in the base-ten representation of $h(n)$. Furthermore, let $k(n)$ be the sum of the squares of the digits of the base-twelve representation of $j(n)$. For example, $h(305) = h(1335_6) = 12 = 12_{10}$, and $j(305) = \text{digit sum of } 12_{10} = 3$, $k(305) = \text{sum of the squares of the digits of } 3_{12} = 9$. Let $M$ be the least value of $n$ such that the base-twenty representation of $k(n)$ cannot be expressed using only the digits $0$ through $9$. Find $M$ modulo $500$.

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