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Another hint
Here’s an example of a sequence generated using only the first 100 integers and the “something involving base 2 (binary) numbers” part of the algorithm used to generate the puzzle sequence:
1
1
2 1
2 2 3 1
2 2 3 2 3 3 4 1
2 2 3 2 3 3 4 2 3 3 4 3 4 4 5 1
2 2 3 2 3 3 4 2 3 3 4 3 4 4 5 2 3 3 4 3 4 4 5 3 4 4 5 4 5 5 6 1
2 2 3 2 3 3 4 2 3 3 4 3 4 4 5 2 3 3 4 3 4 4 5 3 4 4 5 4 5 5 6 2 3 3 4 3
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