ICPC North America Championship 2021

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2021-08-14 07:00 AKDT

ICPC North America Championship 2021

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2021-08-14 10:30 AKDT
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Problem E
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.


Input

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

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 $998\, 244\, 353$.

Sample Input 1 Sample Output 1
ababa
3
Sample Input 2 Sample Output 2
?x?z
703