{"raw_statement":[{"iden":"problem statement","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 $998244353$."},{"iden":"constraints","content":"*   All values in input are integers.\n*   $1 \\leq N \\leq 200$\n*   $1 \\leq a_i \\leq 200$"},{"iden":"input","content":"Input is given from Standard Input in the following format:\n\n$N$\n$a_1$ $a_2$ $\\cdots$ $a_N$"},{"iden":"sample input 1","content":"1\n3"},{"iden":"sample output 1","content":"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 $0$."},{"iden":"sample input 2","content":"2\n5 10"},{"iden":"sample output 2","content":"47"},{"iden":"sample input 3","content":"20\n200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200"},{"iden":"sample output 3","content":"273710435\n\n*   Be sure to find the count modulo $998244353$."}],"translated_statement":null,"sample_group":[],"show_order":["default"],"formal_statement":null,"simple_statement":null,"has_page_source":true}