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I have submitted this question multiple times but received unsatisfactory answers. If you cannot provide a complete and correct solution, please forward it to someone who can. I require a fully worked$-$out solution written neatly by hand, without unnecessary explanations and without AI involvement. The solution must be expert$-$level, as I will evaluate and rate it based on its accuracy and quality. Ensure everything is double$-$checked for correctness before submitting. Thank you!

Question: $($see the image provided$).$