F. Bracket Xoringtime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given a binary string s of length 2n where each element is \mathtt{0} or \mathtt{1}. You can do the following operation: Choose a balanced bracket sequence^\dagger b of length 2n.  For every index i from 1 to 2n in order, where b_i is an open bracket, let p_i denote the minimum index such that b[i,p_i] is a balanced bracket sequence. Then, we perform a range toggle operation^\ddagger from i to p_i on s. Note that since a balanced bracket sequence of length 2n will have n open brackets, we will do n range toggle operations on s. Your task is to find a sequence of no more than 10 operations that changes all elements of s to \mathtt{0}, or determine that it is impossible to do so. Note that you do not have to minimize the number of operations.Under the given constraints, it can be proven that if it is possible to change all elements of s to \mathtt{0}, there exists a way that requires no more than 10 operations.^\dagger A sequence of brackets is called balanced if one can turn it into a valid math expression by adding characters + and 1. For example, sequences "(())()", "()", and "(()(()))" are balanced, while ")(", "(()", and "(()))(" are not.^\ddagger If we perform a range toggle operation from l to r on a binary string s, then we toggle all values of s_i such that l \leq i \leq r. If s_i is toggled, we will set s_i := \mathtt{0} if s_i = \mathtt{1} or vice versa. For example, if s=\mathtt{1000101} and we perform a range toggle operation from 3 to 5, s will be changed to s=\mathtt{1011001}.InputEach test contains multiple test cases. The first line contains the number of test cases t (1 \le t \le 1000). The description of the test cases follows.The first line of each test case contains a single integer n (1 \le n \le 2\cdot 10^5) — where 2n is the length of string s.The second line of each test case contains a binary string s of length 2n (s_i = \mathtt{0} or s_i = \mathtt{1}).It is guaranteed that the sum of n over all test cases does not exceed 2\cdot 10^5.OutputFor each test case, output -1 in a single line if it is impossible to change all elements of s to \mathtt{0}.Otherwise, output a single integer k (0 \le k \le 10) representing the number of operations needed to change all elements of s to \mathtt{0}. Then, on each of the next k lines, output a balanced bracket sequence of length 2n representing the operations needed to change all elements of s to 0s.If there are multiple ways to change all elements of s to \mathtt{0} that require not more than 10 operations, you can output any of them.ExampleInput
4101200003100111401011100Output
-1
2
()()
()()
1
(())()
2
(((())))
()()(())
NoteIn the first test case, it can be proven that it is impossible to change all elements of s to \mathtt{0}.In the second test case, the first operation using the bracket sequence b = \mathtt{()()} will convert the binary string s=\mathtt{0000} to s=\mathtt{1111}. Then, the second operation using the same bracket sequence b = \mathtt{()()} will convert the binary string s=\mathtt{1111} back to s=\mathtt{0000}. Note that since all elements of s is already \mathtt{0} initially, using 0 operations is also a valid answer.In the third test case, a single operation using the bracket sequence b = \mathtt{(())()} will change all elements of s to \mathtt{0}. The operation will happen as follows.  b_1 is an open bracket and p_1 = 4 since b[1,4]=\mathtt{(())} is a balanced bracket sequence. Hence, we do a range toggle operation from 1 to 4 on the binary string s = \mathtt{100111} to obtain s = \mathtt{011011}.  b_2 is an open bracket and p_2 = 3 since b[2,3]=\mathtt{()} is a balanced bracket sequence. Hence, we do a range toggle operation from 2 to 3 on the binary string s = \mathtt{011011} to obtain s = \mathtt{000011}.  b_3 is not an open bracket, so no range toggle operation is done at this step.  b_4 is not an open bracket, so no range toggle operation is done at this step.  b_5 is an open bracket and p_5 = 6 since b[5,6]=\mathtt{()} is a balanced bracket sequence. Hence, we do a range toggle operation from 5 to 6 on the binary string s = \mathtt{000011} to obtain s = \mathtt{000000}.  b_6 is not an open bracket, so no range toggle operation is done at this step. In the fourth test case, the first operation using the bracket sequence b = \mathtt{(((())))} will convert the binary string s = \mathtt{01011100} to s = \mathtt{11111001}. Then, the second operation using the bracket sequence b = \mathtt{()()(())} will convert the binary string s = \mathtt{11111001} to s=\mathtt{00000000}.
4101200003100111401011100
-1 2 ()() ()() 1 (())() 2 (((()))) ()()(()) 
['constructive algorithms', 'greedy', 'implementation', 'math', '*2600']