Unfair Game

AtCoder

IDarc191_e

Time2000ms

Memory256MB

Difficulty—

You are given positive integers $N$, $X$, and $Y$, and two length-$N$ sequences of non-negative integers $A = (A_1,A_2,\ldots,A_N)$ and $B = (B_1,B_2,\ldots,B_N)$. There are $N$ bags, numbered from $1$ to $N$. Initially, bag $i$ contains $A_i$ gold coins and $B_i$ silver coins. Consider the following game played by Takahashi and Aoki using these $N$ bags. First, Takahashi takes some of these bags (possibly zero), and Aoki takes all remaining bags. Then, starting with Takahashi, the two players alternate performing the following operation. * Choose one bag held by the player with at least one gold coin or silver coin in it, and do one of the following two actions for that bag. * Remove one gold coin and add silver coins; the number of silver coins to be added is $X$ for Takahashi and $Y$ for Aoki. This action can only be done if the chosen bag has at least one gold coin. * Remove one silver coin. This action can only be done if the chosen bag has at least one silver coin. * Then, give the chosen bag to the other player. The player who cannot perform the operation loses the game. Find the number, modulo $998244353$, of ways Takahashi can initially take bags so that he will win under optimal play by both players. ## Constraints * $1 \le N \le 2\times 10^5$ * $1 \le X, Y \le 10^9$ * $0 \le A_i, B_i \le 10^9$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $X$ $Y$ $A_1$ $B_1$ $A_2$ $B_2$ $\vdots$ $A_N$ $B_N$ [samples]

Samples

Input #1

2 1 1
1 0
1 1

Output #1

2

Consider the case where Takahashi initially takes bags $1$ and $2$. One possible progression of the game is as follows:

1.  Takahashi chooses bag $2$, removes $1$ gold coin, and adds $1$ silver coin. Then, he gives bag $2$ to Aoki.
    *   Takahashi holds bag $1$ with $1$ gold coin. Aoki holds bag $2$ with $2$ silver coins.
2.  Aoki chooses bag $2$ and removes $1$ silver coin. Then, gives bag $2$ to Takahashi.
    *   Takahashi holds bags $1$ with $1$ gold coin and bag $2$ with $1$ silver coin. Aoki holds none.
3.  Takahashi chooses bag $1$, removes $1$ gold coin, and adds $1$ silver coin. Then, he gives bag $1$ to Aoki.
    *   Takahashi holds bag $2$ with $1$ silver coin. Aoki holds bag $1$ with $1$ silver coin.
4.  Aoki chooses bag $1$, removes $1$ silver coin. Then, he gives bag $1$ to Takahashi.
    *   Takahashi holds bag $1$ which is empty and bag $2$ with $1$ silver coin. Aoki holds none.
5.  Takahashi chooses bag $2$ and removes $1$ silver coin. Then, he gives bag $2$ to Aoki.
    *   Takahashi holds bag $1$ which is empty. Aoki holds bag $2$ which is empty.
6.  Aoki cannot perform the operation, so Aoki loses and Takahashi wins.

Takahashi can win if he initially takes only bag $2$, or if he takes both bags $1$ and $2$. Therefore, the answer is $2$.

Input #2

2 2 1
1 2
1 2

Output #2

3

Takahashi wins if he initially takes bag $1$, bag $2$, or both.

Input #3

Output #3

0

No matter how Takahashi chooses the bags initially, he will lose.

Input #4

Output #4

API Response (JSON)

{
  "problem": {
    "name": "Unfair Game",
    "description": {
      "content": "You are given positive integers $N$, $X$, and $Y$, and two length-$N$ sequences of non-negative integers $A = (A_1,A_2,\\ldots,A_N)$ and $B = (B_1,B_2,\\ldots,B_N)$. There are $N$ bags, numbered from $1",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc191_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given positive integers $N$, $X$, and $Y$, and two length-$N$ sequences of non-negative integers $A = (A_1,A_2,\\ldots,A_N)$ and $B = (B_1,B_2,\\ldots,B_N)$.\nThere are $N$ bags, numbered from $1...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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