2020 AIME II Problem 5

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{4}}) = 10 = 12_{\text{8}}, and g(2020) = \text{the digit sum of }12_{\text{8}} = 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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.