Enumerating Brackets

Enumerating Brackets

A balanced bracket sequence is a string consisting only of the characters “(” (opening brackets) and “)” (closing brackets) such that each opening bracket has a “matching” closing bracket, and vice versa. For example, “(())()” is a balanced bracket sequence, whereas “(())(()” and “())(()” are not.

Given two bracket sequences A and B of the same length, we say that A is lexicographically smaller than B (and write A < B) if:

1. A and B differ in at least one position, and
2. A has a “(”, and B has a “)” in the left-most position in which A and B differ

For example “(())()” < “()()()” because they first differ in the second position from the left, and the first string has an “(” in that position, whereas the second string has a “)”. For a given length N, the “<” operator defines an ordering on all balanced bracket sequences of length N. For example, the ordering of the sequences of length 6 is:

1. ((()))
2. (()())
3. (())()
4. ()(())
5. ()()()

Given a length N and a positive integer M, your task is to find the M-th balanced bracket sequence in the ordering.

Input

You will be given an even integer N, and a positive integer M. It is guaranteed that M will be no more than the number of balanced bracket sequences of length N.

Output

Output the M-th balanced bracket sequence of length N, when ordered lexicographically.

Limits

• 2 ≤ N ≤ 2000
• 1 ≤ M ≤ 10¹⁸
• M ≤ number of balanced bracket sequences of length N

Sample Input 1

6 4

Sample Output 1

()(())