Fascinating puzzle. Every time I thought about it again I found one more cheese. The only unfortunate thing was va gur fbyhgvba V sbhaq, lbh qvqa'g arrq gb renfr n ohggba bapr be hfr gjb fryrpgvbaf ng bapr. Though I did enjoy thinking about them and discarding them as options!
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Lrnu, obgu bs gubfr vqrnf bayl trg hfrq va gur "vagraqrq" (aba-purrfr) fbyhgvba sbe rnpu ebbz. V pbhyq znlor unir svtherq bhg ubj gb fdhrrmr va n ohggba qryrgvba fbzrubj, ohg gur erdhvevat zhygv-fryrpg vf npghnyyl ernyyl gevpxl gb sbepr (rfcrpvnyyl vs gur tbny vf gb erdhver 100% purrfr pbclvat rssvpvrapl, juvpu V yvxr nf n jnl sbe lbh jbex bhg jurer lbh fgvyy arrq gb bcgvzvmr)