Bugfix:

my $valid_re = qr/^(?:$valid)$/;
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Without the grouping there will be strange things happening ;-). Apart from that, depending on the situation, you might want to replace $ at the end of the regex with \z depending on whether you want to allow newlines behind the valid items (your solution) or not (\z).

Update: To see the strange things happening, try

my @valid = qw/foo bar baz/;
my @a_bunch_of_inputs = qw/fooZZZ ZZbarZZ ZZZbaz foobaz foo bar baz/;
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and all of the inputs will produce okay.