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How to check if the ratio is given: $a^{m}b^{n}a^n \rightarrow a^{m}b^{n}a^{m}$ by finite transformations?

I tried using the pumping lemma for finite transducers, but that didn't get me anywhere. Which way should I think?

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    $\begingroup$ I don't understand what you mean by "the ratio is given", or what you mean by "this ratio isn't the finite transformation". I encourage you to edit the question to elaborate on what you are asking. $\endgroup$ – D.W. Dec 19 '20 at 6:45

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