Examine the formal definition of a Turing machine to answer the following questions, and explain your reasoning.
- Can a Turing machine ever write the blank symbol $\sqcup$ on its tape?
- Can the tape alphabet $\Gamma$ be the same as the input alphabet $\Sigma$?
- Can a Turing machine’s head ever be in the same location in two successive steps?
- Can a Turing machine contain just a single state?