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my @list = 
    map {substr($_, 1)}
    sort
    map {$lookup{$_}{val} > 0 ? "A$_" : "B$_"} @list;

sort {
    my $lhs = $lookup{$a}{val} > 0 ? "A$a" : "B$a";
    my $rhs = $lookup{$b}{val} > 0 ? "A$b" : "B$b";
    $lhs cmp $rhs
}