Tournament Construction

一个竞赛图的度数集合是由该竞赛图中每个点的出度所构成的集合。 现给定一个 m 个元素的集合，第 i 个元素是 ai。判断其是否是一个竞赛图的度数 集合,如果是,找到点数最小的满足条件的竞赛图。 $1\le m\le 31$，$0\le a_i\le 30$，$a_i$ 互不相同。

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&lt;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.

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.

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.

2
1 2

4
0011
1001
0100
0010

2
0 3

6
000111
100011
110001
011001
001101
000000