{"raw_statement":[{"iden":"problem statement","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 integers, and $A \\equiv B \\equiv C \\pmod{2}$.\nAt 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.)\nFind the probability, modulo $998244353$, that it is at time $T$ that the three people are at the same point for the first time.\nWhat 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 $R$."},{"iden":"constraints","content":"*   $0 \\leq A, B, C, T \\leq 10^5$\n*   $A \\equiv B \\equiv C \\pmod{2}$\n*   $A, B, C$, and $T$ are integers."},{"iden":"input","content":"The input is given from Standard Input in the following format:\n\n$A$ $B$ $C$ $T$"},{"iden":"sample input 1","content":"1 1 3 1"},{"iden":"sample output 1","content":"873463809\n\nThe 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."},{"iden":"sample input 2","content":"0 0 0 0"},{"iden":"sample output 2","content":"1\n\nThe three people may already be at the same point at time $0$."},{"iden":"sample input 3","content":"0 2 8 9"},{"iden":"sample output 3","content":"744570476"},{"iden":"sample input 4","content":"47717 21993 74147 76720"},{"iden":"sample output 4","content":"844927176"}],"translated_statement":null,"sample_group":[],"show_order":["default"],"formal_statement":null,"simple_statement":null,"has_page_source":true}