I can't think of a good way to also prevent that kind of solutions and ended up prompting "please don't enumerate and filter". Opus 4.7 then reimplemented and proved the backtracking algorithm.

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In the first attempt, Opus 4.7 successfully completed it and proved it. However, the solution is to enumerate all possible space and .filter out anything that doesn't qualify my definition, which is obviously not ideal (if cheating is too harsh).