Consider the following specification. The inputs are Turing machines over input alphabet $=\{$a$,$ b$\}.($a$)$ If the input is a Turing machine M that accepts some input, the output should be any string x that M accepts. $($b$)$ If the input is a Turing machine M that does not accept any input, the output should be any string x$.($There still must be an output, ie$.$ the machine satisfying this specification must halt.$)$ Prove or disprove whether there exists a Turing Machine that halts on every input and satisfies this specification.