RNG and XOR

AtCoder

IDagc034_f

Time3000ms

Memory256MB

Difficulty—

Snuke found a random number generator. It generates an integer between $0$ and $2^N-1$ (inclusive). An integer sequence $A_0, A_1, \cdots, A_{2^N-1}$ represents the probability that each of these integers is generated. The integer $i$ ($0 \leq i \leq 2^N-1$) is generated with probability $A_i / S$, where $S = \sum_{i=0}^{2^N-1} A_i$. The process of generating an integer is done independently each time the generator is executed. Snuke has an integer $X$, which is now $0$. He can perform the following operation any number of times: * Generate an integer $v$ with the generator and replace $X$ with $X \oplus v$, where $\oplus$ denotes the bitwise XOR. For each integer $i$ ($0 \leq i \leq 2^N-1$), find the expected number of operations until $X$ becomes $i$, and print it modulo $998244353$. More formally, represent the expected number of operations as an irreducible fraction $P/Q$. Then, there exists a unique integer $R$ such that $R \times Q \equiv P \mod 998244353,\ 0 \leq R < 998244353$, so print this $R$. We can prove that, for every $i$, the expected number of operations until $X$ becomes $i$ is a finite rational number, and its integer representation modulo $998244353$ can be defined. ## Constraints * $1 \leq N \leq 18$ * $1 \leq A_i \leq 1000$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $A_0$ $A_1$ $\cdots$ $A_{2^N-1}$ [samples]

Samples

Input #1

2
1 1 1 1

Output #1

0
4
4
4

$X=0$ after zero operations, so the expected number of operations until $X$ becomes $0$ is $0$.
Also, from any state, the value of $X$ after one operation is $0$, $1$, $2$ or $3$ with equal probability. Thus, the expected numbers of operations until $X$ becomes $1$, $2$ and $3$ are all $4$.

Input #2

2
1 2 1 2

Output #2

0
499122180
4
499122180

The expected numbers of operations until $X$ becomes $0$, $1$, $2$ and $3$ are $0$, $7/2$, $4$ and $7/2$, respectively.

Input #3

4
337 780 799 10 796 875 331 223 941 67 148 483 390 565 116 355

Output #3

API Response (JSON)

{
  "problem": {
    "name": "RNG and XOR",
    "description": {
      "content": "Snuke found a random number generator. It generates an integer between $0$ and $2^N-1$ (inclusive). An integer sequence $A_0, A_1, \\cdots, A_{2^N-1}$ represents the probability that each of these inte",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc034_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Snuke found a random number generator. It generates an integer between $0$ and $2^N-1$ (inclusive). An integer sequence $A_0, A_1, \\cdots, A_{2^N-1}$ represents the probability that each of these inte...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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