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Contest: Task: Related: TaskD TaskF

Score : $500$ points

Problem Statement

Solve the following problem for $T$ test cases.

Given an integer $N$ and a string $S$, find the number of strings $X$ that satisfy all of the conditions below, modulo $998244353$.

  • $X$ is a string of length $N$ consisting of uppercase English letters.
  • $X$ is a palindrome.
  • $X \le S$ in lexicographical order.
    • That is, $X=S$ or $X$ is lexicographically smaller than $S$.

Constraints

  • $1 \le T \le 250000$
  • $N$ is an integer between $1$ and $10^6$ (inclusive).
  • In a single input, the sum of $N$ over the test cases is at most $10^6$.
  • $S$ is a string of length $N$ consisting of uppercase English letters.

Input

Input is given from Standard Input in the following format:

$T$
$\mathrm{case}_1$
$\mathrm{case}_2$
$\vdots$
$\mathrm{case}_T$

Here, $\mathrm{case}_i$ represents the $i$-th test case.

Each test case is in the following format:

$N$
$S$

Output

Print $T$ lines. The $i$-th line should contain the answer for the $i$-th test case as an integer.

Sample Input 1

5
3
AXA
6
ABCZAZ
30
QWERTYUIOPASDFGHJKLZXCVBNMQWER
28
JVIISNEOXHSNEAAENSHXOENSIIVJ
31
KVOHEEMSOZZASHENDIGOJRTJVMVSDWW

Sample Output 1

24
29
212370247
36523399
231364016

This input contains five test cases.

Test case #1:
The $24$ strings satisfying the conditions are AAA$,$ ABA$,$ ACA$,...,$ AXA.

Test case #2:
$S$ may not be a palindrome.

Test case #3:
Be sure to find the count modulo $998244353$.