Jian-Jia is a young boy who loves playing games. When he is asked a question, he prefers playing games rather than answering directly. Jian-Jia met his friend Mei-Yu and told her about the flight network in Taiwan. There are ~n~ cities in Taiwan (numbered ~0, \dots, n-1~), some of which are connected by flights. Each flight connects two cities and can be taken in both directions.

Mei-Yu asked Jian-Jia whether it is possible to go between any two cities by plane (either directly or indirectly). Jian-Jia did not want to reveal the answer, but instead suggested to play a game. Mei-Yu can ask him questions of the form "Are cities ~x~ and ~y~ directly connected with a flight?", and Jian-Jia will answer such questions immediately. Mei-Yu will ask about every pair of cities exactly once, giving ~r = \frac{n(n-1)} 2~ questions in total. Mei-Yu wins the game if, after obtaining the answers to the first ~i~ questions for some ~i < r~, she can infer whether or not it is possible to travel between every pair of cities by flights (either directly or indirectly). Otherwise, that is, if she needs all ~r~ questions, then the winner is Jian-Jia.

In order for the game to be more fun for Jian-Jia, the friends agreed that he may forget about the real Taiwanese flight network, and invent the network as the game progresses, choosing his answers based on Mei-Yu's previous questions. Your task is to help Jian-Jia win the game, by deciding how he should answer the questions.

Examples

We explain the game rules with three examples. Each example has ~n = 4~ cities and ~r = 6~ rounds of question and answer.

In the first example (the following table), Jian-Jia loses because after round 4, Mei-Yu knows for certain that one can travel between any two cities by flights, no matter how Jian-Jia answers questions 5 or 6.

In the next example, Mei-Yu can prove after round 3 that no matter how Jian-Jia answers questions 4, 5, or 6, one cannot travel between cities 0 and 1 by flights, so Jian-Jia loses again.

In the final example Mei-Yu cannot determine whether one can travel between any two cities by flights until all six questions are answered, so Jian-Jia wins the game. Specifically, because Jian-Jia answered yes to the last question (in the following table), then it is possible to travel between any pair of cities. However, if Jian-Jia had answered no to the last question instead then it would be impossible.

Task

Please write a program that helps Jian-Jia win the game. Note that neither Mei-Yu nor Jian-Jia knows the strategy of each other. Mei-Yu can ask about pairs of cities in any order, and Jian-Jia must answer them immediately without knowing the future questions. You need to implement the following two functions.

• initialize(n) – We will call your initialize first. The parameter ~n~ is the number of cities.
• hasEdge(u, v) – Then we will call hasEdge for ~r = \frac{n(n-1)}{2}~ times. These calls represent Mei-Yu's questions, in the order that she asks them. You must answer whether there is a direct flight between cities ~u~ and ~v~. Specifically, the return value should be 1 if there is a direct flight, or 0 otherwise.

Subtasks

Each subtask consists of several games. You will only get points for a subtask if your program wins all of the games for Jian-Jia.

subtask	points	~n~
1	15	~n = 4~
2	27	~4 \le n \le 80~
3	58	~4 \le n \le 1\,500~

Implementation details

Your submission implements the subprograms described above using the following signatures.

void initialize(int n);
int hasEdge(int u, int v);