But I thought 'ssaa' matches the '$x$y' pattern, where $x is 'ss' and $y is 'aa', that still fits the requirement?

Thanks Abigail-II, I have realized my mistake. Here's an updated version that works -

#!/usr/local/bin/perl -w
use strict;

my @words = qw/ beriberi
    coco
    couscous
    deed
    toot
    toto
    tutu
    assa
    ssaa
    /;

for (@words) {
    print "$_\n" if /^(\w+)\1$/ || /^(\w+)(\w+)\2\1$/;
}
[download]

And output -

beriberi
coco
couscous
deed
toot
toto
tutu
assa
[download]