
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-four representation of  $n$  and let  $g(n)$  be the sum of the digits in the base-eight representation of  $f(n)$ . For example,  $f(2020) = f(133210_\text{four}) = 10 = 12_\text{eight}$ , and  $g(2020) = \text{the digit sum of } 12_\text{eight} = 3$ . Let  $N$  be the least value of  $n$  such that the base-sixteen representation of  $g(n)$  cannot be expressed using only the digits  $0$  through  $9.$  Find the remainder when  $N$  is divided by  $1000.$ 

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