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for my $regexp (qr/^((a*)b|a*b?d)*c\2$/, qr/^(a*b?d|(a*)b)*c\2$/) {
    for my $string (qw( aabadcaa aababdcaa )) {
        say $string =~ /$regexp/ ? 'Yes' : 'No';
    }
}

Yes
No
Yes
Yes

use re 'debug';