{"problem":{"name":"Modulo Nim","description":{"content":"Snuke found a blackboard with nothing written on it. He will do $N$ operations on this blackboard. In the $i$\\-th operation, he chooses an integer between $1$ and $a_i$ (inclusive) and writes it on th","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":6000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"arc134_e"},"statements":[{"statement_type":"Markdown","content":"Snuke found a blackboard with nothing written on it.\nHe will do $N$ operations on this blackboard. In the $i$\\-th operation, he chooses an integer between $1$ and $a_i$ (inclusive) and writes it on the blackboard.\nAfter $N$ integers are written on the blackboard, Taro The First and Jiro The Second will play a game using it. In the game, the two alternately do the operation below, with Taro The First going first.\n\n*   Let $X$ be the largest integer written on the blackboard.\n    *   If $X=0$, the current player **wins** and the game ends.\n*   Choose an integer $m$ between $1$ and $X$ (inclusive).\n*   Replace each of the $N$ integers written on the blackboard with its remainder when divided by $m$.\n\nThere are $\\prod_{i=1}^{N}a_i$ ways for Snuke to write numbers on the blackboard. Find the number of ways among them that will result in Taro The First's win if both players play optimally, modulo $998244353$.\n\n## Constraints\n\n*   All values in input are integers.\n*   $1 \\leq N \\leq 200$\n*   $1 \\leq a_i \\leq 200$\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$\n$a_1$ $a_2$ $\\cdots$ $a_N$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"arc134_e","tags":[],"sample_group":[["1\n3","1\n\n*   Taro The First will win only if Snuke writes $3$.\n*   Otherwise, Taro The First can only play a move that makes the integer on the blackboard $0$."],["2\n5 10","47"],["20\n200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200","273710435\n\n*   Be sure to find the count modulo $998244353$."]],"created_at":"2026-03-03 11:01:14"}}