Random Student ID

AtCoder

IDabc268_g

Time3000ms

Memory256MB

Difficulty—

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). What is the lexicographical order?A string $S = S_1S_2\ldots S_{|S|}$ is said to be **lexicographically smaller** than a string $T = T_1T_2\ldots T_{|T|}$ if one of the following 1. and 2. holds. Here, $|S|$ and $|T|$ denote the lengths of $S$ and $T$, respectively. 1. $|S| \lt |T|$ and $S_1S_2\ldots S_{|S|} = T_1T_2\ldots T_{|S|}$. 2. There exists an integer $1 \leq i \leq \min\lbrace |S|, |T| \rbrace$ satisfying the following two conditions: * $S_1S_2\ldots S_{i-1} = T_1T_2\ldots T_{i-1}$ * $S_i$ is a smaller character than $T_i$. ## 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: $N$ $S_1$ $S_2$ $\vdots$ $S_N$ [samples] ## 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 $R$.

Samples

Input #1

3
a
aa
ab

Output #1

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.

Input #2

3
a
aa
aaa

Output #2

1
2
3

API Response (JSON)

{
  "problem": {
    "name": "Random Student ID",
    "description": {
      "content": "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 ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc268_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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 ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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