CF850D.Tournament Construction

Ivan is reading a book about tournaments. He knows that a tournament is an oriented graph with exactly one oriented edge between each pair of vertices. The score of a vertex is the number of edges going outside this vertex.

Yesterday Ivan learned Landau's criterion: there is tournament with scores _d_1 ≤ _d_2 ≤ ... ≤ d__n if and only if for all 1 ≤ k < n and .

Now, Ivan wanna solve following problem: given a set of numbers S = {_a_1, _a_2, ..., a__m}, is there a tournament with given set of scores? I.e. is there tournament with sequence of scores _d_1, _d_2, ..., d__n such that if we remove duplicates in scores, we obtain the required set {_a_1, _a_2, ..., a__m}?

Find a tournament with minimum possible number of vertices.

伊万正在阅读一本关于竞赛图的书。他了解到，竞赛图是一种有向图，其任意两个顶点之间恰好存在一条有向边。一个顶点的得分是指从该顶点出发的有向边的数量。

昨天，伊万学习了兰道准则（Landau’s criterion）：存在一个得分序列为 $d_1 \le d_2 \le \dots \le d_n$ 的竞赛图，当且仅当对所有 $1 \le k < n$ 满足
，
且满足
。

现在，伊万想解决如下问题：给定一个数集 $S = \{a_1, a_2, \dots, a_m\}$，是否存在一个竞赛图，其得分集合（即所有不同得分构成的集合）恰好为 $S$？换言之，是否存在某个竞赛图，其得分序列为 $d_1, d_2, \dots, d_n$，使得将该序列中重复的得分去除后，所得集合恰好为 $\{a_1, a_2, \dots, a_m\}$？

请找出满足条件的顶点数最少的竞赛图。

The first line contains a single integer m (1 ≤ m ≤ 31).

The next line contains m distinct integers _a_1, _a_2, ..., a__m (0 ≤ a__i ≤ 30) — elements of the set S. It is guaranteed that all elements of the set are distinct.

第一行包含一个整数 $m$（$1 \leq m \leq 31$）。

下一行包含 $m$ 个互不相同的整数 $a_1,\,a_2,\,...,\,a_m$（$0 \leq a_i \leq 30$）——集合 $S$ 的元素。保证集合中所有元素互不相同。

If there are no such tournaments, print string "=(" (without quotes).

Otherwise, print an integer n — the number of vertices in the tournament.

Then print n lines with n characters — matrix of the tournament. The j-th element in the i-th row should be 1 if the edge between the i-th and the j-th vertices is oriented towards the j-th vertex, and 0 otherwise. The main diagonal should contain only zeros.

如果不存在这样的竞赛图，请输出字符串 \=(（不带引号）。

否则，输出一个整数 $n$ —— 竞赛图的顶点数。

然后输出 $n$ 行，每行 $n$ 个字符 —— 竞赛图的邻接矩阵。第 $i$ 行第 $j$ 列的元素应为 $1$，当且仅当从第 $i$ 个顶点指向第 $j$ 个顶点存在一条有向边；否则为 $0$。主对角线上的元素必须全为 $0$。