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By a correct bracketing we mean a string consisting of brackets
), in which the number of opening brackets is equal to the number of closing brackets and each prefix of the string contains at least as many opening brackets, as closing brackets. This way,
()() is a correct bracketing, while
())( is not, because prefix
()) contains more closing brackets than opening brackets.
For a given pair of bracketings of length 2n, we say that the earlier bracketing is the one which has an opening bracket in the first position, where these bracketings differ. Such an ordering is equivalent to lexicographical ordering, assuming that '(' < ')'.
Write a program which:
Input contains exactly two integers n and k (1 ≤ n ≤ 4 000, 1 ≤ k ≤ 1018), separated by a single space.
In the first and only line your program should output a bracketing of length 2n, which is the k-th bracketing among all correct bracketings of length 2n. Input data is prepared in such a way that the bracketing your program is supposed to find always exists.