$($i$)$ When asked to give a recursive definition for the language PALINDROME over the alphabet $\backslash$Sigma $=\{$a$,$ b$\},$ a student wrote:

Rule $1$: a and b are in PALINDROME.

Rule $2$: If x is in PALINDROME, then so are axa and bxb$.$

Unfortunately, all the words in the language defined above have an odd length and so it is not all of PALINDROME. Fix this problem.

$($ii$)$ Give a recursive definition for the language EVENPALINDROME of all palindromes of even length.