Score : $600$ points

Problem Statement

Takahashi Elementary School has $N$ new students. For $i = 1, 2, \ldots, N$, the name of the $i$-th new student is $S_i$ (which is a string consisting of lowercase English letters). The names of the $N$ new students are distinct.

The $N$ students will be assigned a student ID $1, 2, 3, \ldots, N$ in ascending lexicographical order of their names. However, instead of the ordinary order of lowercase English letters where a is the minimum and z is the maximum, we use the following order:

• First, Principal Takahashi chooses a string $P$ from the $26!$ permutations of the string abcdefghijklmnopqrstuvwxyz of length $26$, uniformly at random.
• The lowercase English characters that occur earlier in $P$ are considered smaller.

For each of the $N$ students, find the expected value, modulo $998244353$, of the student ID assigned (see Notes).

Notes

We can prove that the sought expected value is always a rational number. Moreover, under the Constraints of this problem, when the value is represented as $\frac{P}{Q}$ by two coprime integers $P$ and $Q$, we can prove that there is a unique integer $R$ such that $R \times Q \equiv P\pmod{998244353}$ and $0 \leq R \lt 998244353$. Find such $R$.

Constraints

• $2 \leq N$
• $N$ is an integer.
• $S_i$ is a string of length at least $1$ consisting of lowercase English letters.
• The sum of lengths of the given strings is at most $5 \times 10^5$.
• $i \neq j \Rightarrow S_i \neq S_j$

Input

Input is given from Standard Input in the following format:

$N$

$S_1$

$S_2$

$\vdots$

$S_N$

Output

Print $N$ lines. For each $i = 1, 2, \ldots, N$, the $i$-th line should contain the expected value, modulo $998244353$, of the student ID assigned to Student $i$.

3
a
aa
ab

1
499122179
499122179

The expected value of the student ID assigned to Student $1$ is $1$; the expected values of the student ID assigned to Student $2$ and $3$ are $\frac{5}{2}$.

Note that the answer should be printed modulo $998244353$. For example, the sought expected value for Student $2$ and $3$ is $\frac{5}{2}$, and we have $2 \times 499122179 \equiv 5\pmod{998244353}$, so $499122179$ should be printed.

3
a
aa
aaa

1
2
3