qr/ (?(DEFINE)
          ( 00 (?1) | 11 (?2) | )               # group 1:  carry = 0
          ( 01 (?1) | 10 (?3) )                 # group 2:  carry = 1
          ( 00 (?2) | 11 (?3) )                 # group 3:  carry = 2
        )

        \A(?1)\Z /x;