Ex - Trio

AtCoder

IDabc289_h

Time2000ms

Memory256MB

Difficulty—

On a number line are person $1$, person $2$, and person $3$. At time $0$, person $1$ is at point $A$, person $2$ is at point $B$, and person $3$ is at point $C$. Here, $A$, $B$, and $C$ are all integers, and $A \equiv B \equiv C \pmod{2}$. At time $0$, the three people start random walks. Specifically, a person that is at point $x$ at time $t$ ($t$ is a non-negative integer) moves to point $(x-1)$ or point $(x+1)$ at time $(t+1)$ with equal probability. (All choices of moves are random and independent.) Find the probability, modulo $998244353$, that it is at time $T$ that the three people are at the same point for the first time. What is rational number modulo $998244353$? We can prove that the sought probability 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 * $0 \leq A, B, C, T \leq 10^5$ * $A \equiv B \equiv C \pmod{2}$ * $A, B, C$, and $T$ are integers. ## Input The input is given from Standard Input in the following format: $A$ $B$ $C$ $T$ [samples]

Samples

Input #1

1 1 3 1

Output #1

873463809

The three people are at the same point for the first time at time $1$ with the probability $\frac{1}{8}$. Since $873463809 \times 8 \equiv 1 \pmod{998244353}$, $873463809$ should be printed.

Input #2

0 0 0 0

Output #2

1

The three people may already be at the same point at time $0$.

Input #3

0 2 8 9

Output #3

744570476

Input #4

47717 21993 74147 76720

Output #4

844927176

API Response (JSON)

{
  "problem": {
    "name": "Ex - Trio",
    "description": {
      "content": "On a number line are person $1$, person $2$, and person $3$. At time $0$, person $1$ is at point $A$, person $2$ is at point $B$, and person $3$ is at point $C$.   Here, $A$, $B$, and $C$ are all inte",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc289_h"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "On a number line are person $1$, person $2$, and person $3$. At time $0$, person $1$ is at point $A$, person $2$ is at point $B$, and person $3$ is at point $C$.  \nHere, $A$, $B$, and $C$ are all inte...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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