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Let us assume a binary alphabet $\Sigma=\{a,b\}.$ Two words $u,v\in \Sigma^{\ast}$ are said to be conjugates if there exist $w_{1},w_{2}\in \Sigma^{\ast}$ such that $u=w_{1}w_{2}$ and  $v=w_{2}w_{1}.$ Prove that $u$ and $v$ are conjugates if and only if there exists  $w\in \Sigma^{\ast}$ such that $uw=wv.$

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