Construct a Turing machine T with tape symbols $1,2,$ and B that, when given an integer string as input, halts at the greatest number of the tape. The cells storing the given string are the only cells of the tape that contain non$-$blank symbols. The control unit of T is initially positioned at the first non$-$blank symbol and can only move left or right. Define a set of states, a starting state, a set of final states, and transition rules.