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    This comment is hidden because it contains spoiler information about the solution

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    Ah, but it is! Can you elaborate on what doesn't look right about it? Maybe it looks "backwards"? I actually had to check it after your comment to make sure it worked correctly, but it does. I wrote a comment in response to @Lavios below to offer a theoretical justification for why it works. Maybe that will help, but if it doesn't, I'd be happy to answer any questions you have.