$\text{Example}:$ Design a Turing machine that copies strings of $1’s$. More precisely, find a machine that performs the computation
$q_0w \vdash^* q_fww,$
for any $w\in\{1\}^+$.
Give the sequence of instantaneous descriptions that the Turing machine in Example goes through when presented with the input $111$. What happens when this machine is started with $110$ on its tape$?$