{"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 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 $R$.\n\n## Constraints\n\n*   $0 \\leq A, B, C, T \\leq 10^5$\n*   $A \\equiv B \\equiv C \\pmod{2}$\n*   $A, B, C$, and $T$ are integers.\n\n## Input\n\nThe input is given from Standard Input in the following format:\n\n$A$ $B$ $C$ $T$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"abc289_h","tags":[],"sample_group":[["1 1 3 1","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."],["0 0 0 0","1\n\nThe three people may already be at the same point at time $0$."],["0 2 8 9","744570476"],["47717 21993 74147 76720","844927176"]],"created_at":"2026-03-03 11:01:14"}}