$1)(20$ points$)($Annual exam $2023)$

$($a$)$ Let A $=\{0,1\}$ be a finite alphabet, and consider the language L

consisting of all strings over the alphabet A that contain exactly

three $1$s$.$ In other words, $10101$ that contains three $1$s is in L$,$

but $0101$ that contains only two $1$s is not. Write down the

algorithm of a Turing machine that recognises L and process

the following strings according to your algorithm: $,01,101,$

$01101,$ and $0110110.$

$($b$)$ Write down the transition diagram of the Turing machine from

part $($a$)$ carefully labelling the initial state, the accept state, the

reject state, and all the transitions specified in your algorithm