For any set of strings A$,$ define the set Triplicated$($A$)=\{$x $$ xxx in A$\}.$ That is$,$ Triplicated$($A$)$ consists of strings where x$^3$ in A$.$ For instance, if A $=\{,01,000,011,001001001\},$ then Triplicated$($A$)=\{,0,001\}.$ Let A be an

arbitrary regular language. Prove that Triplicated$($A$)$ is regular, too. Hint: Suppose you have a DFA M so that L$($M$)=$ A$.$ Suppose $(1)$ you knew which state M would be in after scanning x$,$ and $(2)$ you knew what state M would be in after scanning xx$.$ How could you then check if xxx in A$?$ Then use the power of nondeterminism.