P16017 [ICPC 2021 NAC] Ketek Counting

Define a *Ketek* to be a sentence that reads the same forwards and backwards, by word. For example, ‘fall leaves after leaves fall’ is a *Ketek* since the words in reverse order are the same as the original order. Given a string consisting of lower-case letters and the character ‘?’, count the number of distinct *Keteks* you can make by replacing every ‘?’ with lower-case letters (one letter per ‘?’), and optionally adding spaces between any letters. Note that a *Ketek* cannot contain any ?’s; they all must be replaced exclusively by lower-case letters. For example, if we start with the string ‘ababa’, we can form 3 different *Keteks*: ‘ababa’, ‘a bab a’ and ‘a b a b a’. If we start with the string ‘?x?z’ instead, we can form 703 different *Keteks*: - There are $26^2 = 676$ ways to replace the ?’s and form a one-word *Ketek*. - Add spaces to form ‘? x? z’. There are 26 ways to form a *Ketek* (the first ‘?’ must be z; the other can be any lower-case letter). - Add a space to form ‘?x ?z’. There is no way to form a *Ketek*. - Add spaces to form ‘? x ? z’. There is one way to form a *Ketek* (the first ‘?’ must be z; the second must be x). The total is $676 + 26 + 0 + 1 = 703$. Two *Keteks* are different if they have a different number of words, or there is some word index where the words are not the same.

The single line of input contains a string $s$ ($1 \le |s| \le 30{,}000$), which consists of lower-case letters (‘a’–‘z’) and the character ‘?’.

Output the number of distinct *Keteks* that can be formed by replacing the ?’s with lower-case letters and adding spaces. Since this number may be large, output it modulo $998{,}244{,}353$.