$9.$ Let s be a string of size n $=2$k for some k $<26$ that consists of only lowercase letters. The string s is said to be c$-$beautiful $($c is an arbitrary character$)$ if one of the three conditions is satisfied:

$$ The size of s is $1$ and it consists of the character c$.$

$$ The size of s is greater than $1,$ the first half of s only consists of c $($s$1=$ s$2=...=$ sn$/2=$ c$)$ and the second half of s $($sn$/2+1$sn$/2+2...$sn$)$ is a $($c $+1)-$beautiful string.

$$ The size of s is greater than $1,$ the first half of s $($s$1$s$2...$sn$/2)$ is a $($c $+1)-$beautiful string and the second half of s only consists of c $($sn$/2+1=$ sn$/2+2=...=$ sn $=$ c$).$

$($c $+1)$ denotes the next lowercase letter after c in alphabetical order. For example, $$ccde$$ is $$c$-$beautiful, and $$jjlkiiii$$ is $$i$-$beautiful.

In an operation, you can choose any index in s and change the letter at that index to an arbitrary character. Design an O$($n log n$)$ D&C algorithm to find the minimum number of operations to convert s into an $$a$-$beautiful string.

Hint: When the size of s is $1,$ it$$s the base case. When the size of s is greater than $1,$ there are only two cases, consider those two cases and take the minimum. $(5)$