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$\text{Example}:$ 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$?$
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