{"raw_statement":[{"iden":"problem statement","content":"There are $N$ balls numbered $1$ through $N$. Initially, Ball $i$ is painted in Color $A_i$.\nColors are represented by integers between $1$ and $N$, inclusive.\nConsider repeating the following operation until all the colors of the balls become the same.\n\n*   There are $2^N$ subsets (including the empty set) of the set consisting of the $N$ balls; choose one of the subsets uniformly at random. Let $X_1,X_2,...,X_K$ be the indices of the chosen balls. Next, choose a permutation obtained by choosing $K$ integers out of integers $(1,2,\\dots,N)$ uniformly at random. Let $P=(P_1,P_2,\\dots,P_K)$ be the chosen permutation. For each integer $i$ such that $1 \\le i \\le K$, change the color of Ball $X_i$ to $P_i$.\n\nFind the expected value of number of operations, modulo $998244353$.\nHere a permutation obtained by choosing $K$ integers out of integers $(1,2,\\dots,N)$ is a sequence of $K$ integers between $1$ and $N$, inclusive, whose elements are pairwise distinct."},{"iden":"notes","content":"We can prove that the sought expected value is always a rational number. Also, under the Constraints of this problem, when the value is represented by two coprime integers $P$ and $Q$ as $\\frac{P}{Q}$, we can prove that there exists a unique integer $R$ such that $R \\times Q \\equiv P(\\bmod\\ 998244353)$ and $0 \\le R < 998244353$. Find this $R$."},{"iden":"constraints","content":"*   $2 \\le N \\le 2000$\n*   $1 \\le A_i \\le N$\n*   All values in input are integers."},{"iden":"input","content":"Input is given from Standard Input in the following format:\n\n$N$\n$A_1$ $A_2$ $\\dots$ $A_N$"},{"iden":"sample input 1","content":"2\n1 2"},{"iden":"sample output 1","content":"4\n\nThe operation is repeated until a subset of size $1$ is chosen and the color of the ball is changed to the color of the ball not contained in the subset. The probability is $\\displaystyle \\frac{2}{4} \\times \\frac{1}{2}=\\frac{1}{4}$, so the expected value is $4$."},{"iden":"sample input 2","content":"3\n1 1 1"},{"iden":"sample output 2","content":"0\n\nThe operation is never performed, since the colors of the balls are already the same."},{"iden":"sample input 3","content":"10\n3 1 4 1 5 9 2 6 5 3"},{"iden":"sample output 3","content":"900221128"}],"translated_statement":null,"sample_group":[],"show_order":["default"],"formal_statement":null,"simple_statement":null,"has_page_source":true}