$($a$)(15$ pts$)$ Give the context$-$free grammar for the language $L$ which contains binary numbers such that each number contains odd$-$number of digits, and the first, the last and the middle digits of the number are same. e$.$g$.$ the strings $"1","000","01000","10111,"10100100001"$ are in the language L$,$ whereas $"100","1011","1010011"$ are not.

$1w_n+S_{n}1$

$0w_{n}0i{s}_{n}0$

$($c$)(25$ pts$)$ Prove or disprove that the language $L$ defined in part $($a$)$ above is regular.

PLEASE ONLY C PART SOLUT$$ON