Build a Turing Machine that multiplies two integers. Explain the logic behind your design shortly. An example initial and final configuration is given below.

Initial configuration:aaaaa $bbdots..$

Final configuration: AAAAA $bbccccccccccdots.$

The example above indicates that $5$ is multiplied by $2$ and the result which consists of $10c\text{'}$s resides on the tape after the input. You can change the letters during the process as in the above example a$\text{'}$s are turned to A$\text{'}$s to mark them during the process of the Turing machine.