Design a Turing machine that finds the middle of a string of even length. Specifically, if $w = a_1a_2...a_na_{n+1}...a_{2n},$ with $a_i\in\Sigma,$ the Turing machine should produce $\widehat{w} = a_1a_2...a_nca_{n+1}...a_{2n},$ where $c\in\Gamma-\Sigma$.
