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#!/usr/bin/perl
use warnings;
use strict;
...
for my $string (grep 1 == @{ $seen{$_} }, keys %seen) {
    print "$string unique in $seen{$string}[0]\n";
}

$ perl 1077045.pl [123].txt
cat unique in 1.txt
dog unique in 3.txt
rat unique in 2.txt