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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:
(", 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 Mth balanced bracket sequence in the ordering.
You will be given an even integer N (2 ≤ N ≤ 2000), and a positive integer M. It is guaranteed that M will be no more than 1018 and no more than the number of balanced bracket sequences of length N (whichever is smaller).
Output the Mth balanced bracket sequence of length N, when ordered lexicographically.