{"problem":{"name":"Takahashi And Pass-The-Ball Game","description":{"content":"There are $N$ Takahashi. The $i$\\-th Takahashi has an integer $A_i$ and $B_i$ balls. An integer $x$ between $1$ and $K$, inclusive, will be chosen uniformly at random, and they will repeat the followi","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"abc310_g"},"statements":[{"statement_type":"Markdown","content":"There are $N$ Takahashi.\nThe $i$\\-th Takahashi has an integer $A_i$ and $B_i$ balls.\nAn integer $x$ between $1$ and $K$, inclusive, will be chosen uniformly at random, and they will repeat the following operation $x$ times.\n\n*   For every $i$, the $i$\\-th Takahashi gives all his balls to the $A_i$\\-th Takahashi.\n\nBeware that all $N$ Takahashi simultaneously perform this operation.\nFor each $i=1,2,\\ldots,N$, find the expected value, modulo $998244353$, of the number of balls the $i$\\-th Takahashi has at the end of the operations.\nHow to find a expected value modulo $998244353$ It can be proved that the sought probability is always a rational number. Additionally, the constraints of this problem guarantee that if the sought probability is represented as an irreducible fraction $\\frac{y}{x}$, then $x$ is not divisible by $998244353$. Here, there is a unique $0\\leq z\\lt998244353$ such that $y\\equiv xz\\pmod{998244353}$, so report this $z$.\n\n## Constraints\n\n*   $1\\leq N\\leq 2\\times10^5$\n*   $1\\leq K\\leq 10^{18}$\n*   $K$ is not a multiple of $998244353$.\n*   $1\\leq A _ i\\leq N\\ (1\\leq i\\leq N)$\n*   $0\\leq B _ i\\lt998244353\\ (1\\leq i\\leq N)$\n*   All input values are integers.\n\n## Input\n\nThe input is given from Standard Input in the following format:\n\n$N$ $K$\n$A _ 1$ $A _ 2$ $\\cdots$ $A _ N$\n$B _ 1$ $B _ 2$ $\\cdots$ $B _ N$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"abc310_g","tags":[],"sample_group":[["5 2\n3 1 4 1 5\n1 1 2 3 5","3 0 499122179 499122178 5\n\nDuring two operations, the five Takahashi have the following number of balls.\n![image](https://img.atcoder.jp/abc310/eeca44e66744660173a72967840e158a.png)\nIf $x=1$ is chosen, the five Takahashi have $4,0,1,2,5$ balls.  \nIf $x=2$ is chosen, the five Takahashi have $2,0,4,1,5$ balls.\nThus, the sought expected values are $3,0,\\dfrac52,\\dfrac32,5$. Print these values modulo $998244353$, that is, $3,0,499122179,499122178,5$, separated by spaces."],["3 1000\n1 1 1\n1 10 100","111 0 0\n\nAfter one or more operations, the first Takahashi gets all balls."],["16 1000007\n16 12 6 12 1 8 14 14 5 7 6 5 9 6 10 9\n719092922 77021920 539975779 254719514 967592487 476893866 368936979 465399362 342544824 540338192 42663741 165480608 616996494 16552706 590788849 221462860","817852305 0 0 0 711863206 253280203 896552049 935714838 409506220 592088114 0 413190742 0 363914270 0 14254803"],["24 100000000007\n19 10 19 15 1 20 13 15 8 23 22 16 19 22 2 20 12 19 17 20 16 8 23 6\n944071276 364842194 5376942 671161415 477159272 339665353 176192797 2729865 676292280 249875565 259803120 103398285 466932147 775082441 720192643 535473742 263795756 898670859 476980306 12045411 620291602 593937486 761132791 746546443","918566373 436241503 0 0 0 455245534 0 356196743 0 906000633 0 268983266 21918337 0 733763572 173816039 754920403 0 273067118 205350062 0 566217111 80141532 0"]],"created_at":"2026-03-03 11:01:14"}}