My Login : Adbd$738$

Kindly proceed using loginThe following is required for this question.

Suppose $$ is an alphabet containing the last three letters in your login$.$ If two of these letters are the same, then for the last appearance of the repeated letter you must select the next letter of the English alphabet to ensure your alphabet has three different letters. If all last three letters in your login are the same, then you must select in order of appearance the next two letters of the English alphabet. If two of the last three letters in your login are the same and the next in the alphabet coincides with the third, then select the subsequent letter in the alphabet for the second appearance of the repeated letter. Call these letters $,,,$ in the order of appearance. Next, let i and $j$ be the first and second digits of your login; if any of these is $0,$ then you must add $1$ to it to ensure that both i and $j$ are nonzero $($ i and $j$ can be equal$).$

For example, if your login is abcd$123$ then $=b,=c,=d,i=1,$ and $j=2.$

If your $login$ is acec $113$ then $=c,=e,=d,i=1,$ and $j=1.$

If your $login$ is acdc$123$ then $=c,=d,=e,i=1,$ and $j=2,$

If your $log$ in is accc$123$ then also $=c,=d,=e,i=1,$ and $j=2.$

Consider the following language $L$ :

$L=\{{}^{k(i+1)}{}^{(1+j+m)}{}^{(i+m)}|k0,m0\}.$

Thus, for the example login abcd$123,$ the alphabet is $=\{b,c,d\},$ and the language is

$L=\{b^{2k}c{d}^{(3+m)}{c}^{(1+m)}|k0,m0\}.$

Task:

Write your login$,$ the corresponding alphabet $,$ and the corresponding language $L.$

Is your language regular? If so$,$ give a regular expression for the language. If not, demonstrate that the language is not regular using the Pumping Lemma for regular languages.