Ex - Trio
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Ex - Trio
Score : $600$ points
Problem Statement
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 by two coprime integers and , we can prove that there is a unique integer such that and . Find such .
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:
Output
Find the probability, modulo $998244353$, that it is at time $T$ that the three people are at the same point for the first time, and print the answer.
1 1 3 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.
0 0 0 0
1
The three people may already be at the same point at time $0$.
0 2 8 9
744570476
47717 21993 74147 76720
844927176