Instructions: $1.$ For each language below, construct a PDA and a CFG that accepts the language. $2.$ Provide both an informal description of how the PDA works and a formal definition of the PDA and CFG$.3.$ Where necessary, illustrate the PDA's and CFG$$s behavior by showing how it processes a sample string. $4.$ Ensure that the PDAs are deterministic wherever possible $($nondeterministic PDAs can be used if necessary$).$ Problem $1$: PDA and CFG for the Language L$1=\{$anbnck $$ n$,$ k $0\}$ Language Description: $$ The language consists of strings with an equal number of a$\text{'}$s and b$\text{'}$s$,$ followed by any number of c$\text{'}$s$.$ Example strings: abc, aabbccccc, aaabbb, cc$.$ Problem $2$: PDA and CFG for the Language L$2=\{$w $$ w is a string of balanced square brackets $([])\}$ Language Description: $$ The language consists of balanced square brackets. Every opening bracket $[$ must be matched with a corresponding closing bracket $].$ Example strings: $[],[][],[[]],[[[[]]]].$ Problem $3$: PDA and CFG for the Language L$3=\{$anbm $$ m$$n$0\}$ Language Description: $$ The language consists of strings with any number of a$\text{'}$s followed by at least the same number of b$\text{'}$s $($but possibly more$).$ Example strings: abb, aabbbb, abbb, bb$.$ Problem $4$: PDA and CFG for the Language L$4=\{$w $$ contains an equal number of occurrences of a and b$\}$ Language Description: $$ The language consists of strings where the number of a$\text{'}$s is equal to the number of b$\text{'}$s$,$ regardless of the order of their appearance. $$ Example strings: ab$,$ baab, baba, aabb. $$ Problem$5$: Construct a PDA and CFG for the language L$5=\{$ai bj ck $$ i$=$j or j$=$k$,$ i$,$ j$,$ k $0\}$