$30$ points$)$ You are $($still$)$ working on a fantasy novel. For the dwarves, you have decided to create a language over the alphabet $=\{$k$,$ u$,$ d$,$ z$,$ e$,$ f $\}$ with the following rules: Words that start with k cannot end with an f$.$ u can not come directly after z$,$ k or f$.$ e can only come directly after z$,$ d or u$.$ Words must have more short letters $($us$,$ es$,$ and zs$)$ than tall letters $($ks$,$ ds$,$ and fs$)$ The number of fs and us combined must be the same as the number of zs and es combined.Show that this language is not context$-$free using the pumping lemma for context$-$free languages, by following these steps:$($a$)(5$ points$)$ Construct a valid word w of length at least n $($you do not know what n is$!).($b$)(15$ points$)$ Determine all possible, valid ways, that w can be split into the five parts uxyzv.$($c$)(10$ points$)$ Show that none of these possible splits can be pumped $($if x and z are repeated a certain number of times, or removed entirely, the word will become invalid$).$