E. New Schooltime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou have decided to open a new school. You have already found n teachers and m groups of students. The i-th group of students consists of k_i \geq 2 students. You know age of each teacher and each student. The ages of teachers are a_1, a_2, \ldots, a_n and the ages of students of the i-th group are b_{i, 1}, b_{i, 2}, \ldots, b_{i, k_i}.To start lessons you should assign the teacher to each group of students. Such assignment should satisfy the following requirements:  To each group exactly one teacher assigned.  To each teacher at most 1 group of students assigned.  The average of students' ages in each group doesn't exceed the age of the teacher assigned to this group. The average of set x_1, x_2, \ldots, x_k of k integers is \frac{x_1 + x_2 + \ldots + x_k}{k}.Recently you have heard that one of the students will refuse to study in your school. After this, the size of one group will decrease by 1 while all other groups will remain unchanged.You don't know who will refuse to study. For each student determine if you can start lessons in case of his refusal.Note, that it is not guaranteed that it is possible to start lessons before any refusal.InputThe first line contains a single integer t (1 \leq t \leq 1000) — the number of test cases.The first line of each test case contains two integers n and m (1 \leq m \leq n \leq 10^5) — the number of teachers and the number of groups of students.The second line of each test case contains n integers a_1, a_2, \ldots, a_n (1 \leq a_i \leq 10^5) — the ages of teachers.The next 2m lines contains descriptions of groups. The first line of description of group contains a single integer k_i (2 \leq k_i \leq 10^5) — the number of students in this group. The second line of description of group contains k_i integers b_{i, 1}, b_{i, 2}, \ldots, b_{i, k_i} (1 \leq b_{i, j} \leq 10^5) — the ages of students of this group.It is guaranteed that the total sum of n over all test cases doesn't exceed 10^5 and that the total sum of k_1 + k_2 + \ldots + k_m over all test cases doesn't exceed 2 \cdot 10^5OutputFor each test case output string of symbols 0 and 1 of length k_1 + k_2 + \ldots + k_m. The i-th symbol of this string should be equals 1 if it is possible to start lessons in case of the i-th student refuse and it should be equals 0 otherwise.Students are numbered by integers from 1 to k_1 + k_2 + \ldots + k_m in the order they appear in the input. Thus, students of the 1-st group are numbered by integers 1, 2, \ldots, k_1, students of the 2-nd group are numbered by integers k_1 + 1, k_1 + 2, \ldots, k_1 + k_2 and so on.ExampleInput
2
1 1
30
3
25 16 37
4 2
9 12 12 6
2
4 5
3
111 11 11
Output
101
00100
NoteIn the first test case there is one group of students with average age \frac{25+16+37}{3}=26 and one teacher with age 30. There exists only one assignment that allows to start lessons.If the student with age 16 will refuse to study, the average age of students in this group will become \frac{25+37}{2}=31 so there won't be any assignment that allows to start lessons.In the second test case it is impossible to start lessons initially. However, if the 3-rd student with age 111 will refuse to study, the average ages of groups will become \frac{4 + 5}{2}=4.5 and \frac{11+11}{2} = 11 correspondingly. Then it is possible to assing the first group to the first teacher and the second group to the third teacher.
2
1 1
30
3
25 16 37
4 2
9 12 12 6
2
4 5
3
111 11 11

101 00100 
['binary search', 'data structures', 'dp', 'greedy', 'implementation', 'sortings', '*2300']