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:
- A and B differ in at least one position, and
- 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:
Given a length N and a positive integer M, your task is to find the M-th balanced bracket sequence in the ordering.
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 the M-th balanced bracket sequence of length N, when ordered lexicographically.
- 2 ≤ N ≤ 2000
- 1 ≤ M ≤ 1018
- M ≤ number of balanced bracket sequences of length N
Sample Input 1
Sample Output 1