$1)$ 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$\backslash$Delta bb$\backslash$Delta $\backslash$Delta $\backslash$Delta $....$ Final configuration: AAAAA$\backslash$Delta bb$\backslash$Delta cccccccccc$\backslash$Delta $\backslash$Delta $\backslash$Delta $....$ The example above indicates that $5$ is multiplied by $2$ and the result which consists of $10$ c$\text{'}$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.