Beautiful League

~~见WC2007剪刀石头布~~ 一片美丽的土地上有$n(n<=50)$支足球队，他们将两两进行比赛，共进行$n*(n-1)/2$场比赛（每局比赛都有赢家和输家，没有平局） 比赛结束后，主办方会统计美丽的三队组数量。若队伍$A$胜了队伍$B$，队伍$B$胜了队伍$C$，队伍$C$胜了队伍$A$，则称$(A,B,C)$是美丽的三队组。 （顺序很重要！$(A,B,C)$和$(B,C,A)$是两个不同的三队组） 现在已经开始了$m(m<=n*(n-1)/2)$场比赛，这$m$场比赛的结果是已知的。请问在打完这$n*(n-1)/2$场比赛后，最多有多少组美丽的三队组呢？

A football league has recently begun in Beautiful land. There are $ n $ teams participating in the league. Let's enumerate them with integers from $ 1 $ to $ n $ . There will be played exactly $ \frac{n(n-1)}{2} $ matches: each team will play against all other teams exactly once. In each match, there is always a winner and loser and there is no draw. After all matches are played, the organizers will count the number of beautiful triples. Let's call a triple of three teams $ (A, B, C) $ beautiful if a team $ A $ win against a team $ B $ , a team $ B $ win against a team $ C $ and a team $ C $ win against a team $ A $ . We look only to a triples of different teams and the order of teams in the triple is important. The beauty of the league is the number of beautiful triples. At the moment, $ m $ matches were played and their results are known. What is the maximum beauty of the league that can be, after playing all remaining matches? Also find a possible results for all remaining $ \frac{n(n-1)}{2} - m $ matches, so that the league has this maximum beauty.

The first line contains two integers $ n, m $ ( $ 3 \leq n \leq 50, 0 \leq m \leq \frac{n(n-1)}{2} $ ) — the number of teams in the football league and the number of matches that were played. Each of $ m $ following lines contains two integers $ u $ and $ v $ ( $ 1 \leq u, v \leq n $ , $ u \neq v $ ) denoting that the $ u $ -th team won against the $ v $ -th team. It is guaranteed that each unordered pair of teams appears at most once.

Print $ n $ lines, each line having a string of exactly $ n $ characters. Each character must be either $ 0 $ or $ 1 $ . Let $ a_{ij} $ be the $ j $ -th number in the $ i $ -th line. For all $ 1 \leq i \leq n $ it should be true, that $ a_{ii} = 0 $ . For all pairs of teams $ i \neq j $ the number $ a_{ij} $ indicates the result of the match between the $ i $ -th team and the $ j $ -th team: - If $ a_{ij} $ is $ 1 $ , the $ i $ -th team wins against the $ j $ -th team; - Otherwise the $ j $ -th team wins against the $ i $ -th team; - Also, it should be true, that $ a_{ij} + a_{ji} = 1 $ . Also note that the results of the $ m $ matches that were already played cannot be changed in your league. The beauty of the league in the output should be maximum possible. If there are multiple possible answers with maximum beauty, you can print any of them.

3 1
1 2

010
001
100

4 2
1 2
1 3

0110
0001
0100
1010

The beauty of league in the first test case is equal to $ 3 $ because there exists three beautiful triples: $ (1, 2, 3) $ , $ (2, 3, 1) $ , $ (3, 1, 2) $ . The beauty of league in the second test is equal to $ 6 $ because there exists six beautiful triples: $ (1, 2, 4) $ , $ (2, 4, 1) $ , $ (4, 1, 2) $ , $ (2, 4, 3) $ , $ (4, 3, 2) $ , $ (3, 2, 4) $ .