NACLO 2023 (D) For The Sake of Their House

We do the problem "For the Sake of Their House" for NACLO 2023 (North American Computational Linguistics Olympiad), a competition for high-schoolers in North America.
This a solid problem with a couple inferences that aren't necessarily obvious.
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[@kilaldiwhiteman8436]

Shouldn’t it be kokkeznitlen?