Decider vs Recognizer $(10$ points$)($Graded for correctness$)$ Consider the Turing machine,

$M,$ determined by the following state diagram.

$($a$)$ Give a string of length $3$ that is accepted by $M.$ Explain why it is accepted by tracing

through the computation of $M$ on this string.

$($b$)$ Is $M$ a decider? Prove your answer.

$($c$)$ Prove or disprove the statement:

Every Turing$-$recognizable language can be recognized by a TM that either accepts

or loops, but never rejects.

Illustrate your proof on the TM given in this problem, either by describing the diagram of

a TM with the required property, or by explaining why such TM does not exist.