Automata theory pumping lemmaproblem #1: these are short answer questions concerning the pumping lemma for regular sets. assume that we are in the middle of a pumping lemma proof designed to show that a language is not regular, and that the value "n" and all other preliminaries have been determined. the questions below address specifically the "w" string, chosen in step 2 of the proof, and the "i" value, chosen in step 4 of the proof. a) let l = {0i1j0k| i, j,k > = 0 and i< =3k}i) if w = 0n1n0n, what is the minimum "i" value that will work? give an argument to show that your "i" value is correct.(1) will i=0 work? why or why not? ii) if w = 0n1n02n, what is the minimum "i" value that will work? argument to show that your "i" value is correct.(1) will i=0 work? why or why not? iii) if w = 03n1n0n, what is the minimum "i" value that will work? argument to show that your "i" value is correct. give an argument to show that your "i" value is correct. iv) if w = 02n1n03n, what is the minimum "i" value that will work, and why? v) is w = 03n1n0n an acceptable "w" string? why or why not? b) let l = {0i1j0k| i, j,k > = 0 and j< =2i}i) let w = 0n1n0n(1) is this an acceptable "w" string in step 2 of the proof? why or why not? (2) will it work in step 4? why or why not? ii) what "w" string do you recommend for a pumping lemma proof for this language? (1) choose an "i" value, and give an argument that your choices for "w" and "i" will work in a pumping lemma proof.