$(10\%)$ Given a Turing machine, $Q=\{q_0,q_1,q_2,q_3\},{\displaystyle \underset{?}{\overset{?}{}}}=\{0,1\},=\{0,1,\}$ and $F=\{q_3\},$ with

the following transitions

$(q_0,0)=(q_1,0,R)$

$(q_1,1)=(q_1,1,R)$

$(q_1,0)=(q_2,0,R)$

$(q_2,0)=(q_2,0,R)$

$(q_2,)=(q_3,,R)$

$($a$)$ Show that the Turing machine accepts the string $0000.$

$($b$)$ What language is accepted by the Turing machine?