Let $=\{$a$,$ b$,$ c$\}$ and let K$,$ L $$ be languages described as follows:

K $=\{$amb$2$cn : m$,$ n $2$ N$\}$; L $=\{$anbn : n $2$ Z$0\}.$

$($a$)$ Describe K and L recursively.

Answer : K : ab$2$c $2$ K$,$ x $2$ K $![$ax $2$ K $^$ xc $2$ K$].$

L : ; $2$ L$,$ x $2$ L $!$ axb $2$ L$.$

$($b$)$ Construct a finite state automaton $($FSA$)$ that accepts K$.$