$[20$ Points$]$ Give a Turing machine for each of the following languages:

$($a$)$ L$1=\{$anbmck $|$m$$n$,$k$1\}.$

$($b$)$ L$2=\{$xy $|$ x in $\{$a$,$b$\}+,$y in $\{$c$\}+$ and na$($x$)=$ nc$($y$)\}.$