A polynomial$-$time three$-$outcome NTM M is a polynomial$-$time NTM with three

$($instead of two$)$ possible outcomes: accept, reject, and unsure $($consider them as three special

states qaccept, qreject and qunsure at which M may halt$).$ We say such a machine decides a language

L if the following is true: If x in L$,$ all branches of M end up with accept or unsure, and at least

one with accept; If x $/$ in L$,$ all branches end up with reject or unsure, and at least one with reject.

$($a$)$ Show that if L is decided by a polynomial$-$time three$-$outcome NTM$,$ then both L$,$L in NP$.$

$($Hint: Focus on L in NP$.$ The proof of L in NP is symmetric.$)$

$($b$)$ Show that if both L$,$L in NP$,$ then L is decided by a polynomial$-$time three$-$outcome NTM