Problem:
Score : 400 points

Problem Statement

You are given a string S of length N consisting of ( and ). Your task is to insert some number of ( and ) into S to obtain a correct bracket sequence.

Here, a correct bracket sequence is defined as follows:

() is a correct bracket sequence.

If X is a correct bracket sequence, the concatenation of (, X and ) in this order is also a correct bracket sequence.

If X and Y are correct bracket sequences, the concatenation of X and Y in this order is also a correct bracket sequence.

Every correct bracket sequence can be derived from the rules above.

Find the shortest correct bracket sequence that can be obtained. If there is more than one such sequence, find the lexicographically smallest one.

Constraints

The length of S is N.

1 ≤ N ≤ 100

S consists of ( and ).

Input

Input is given from Standard Input in the following format:

N
S

Output

Print the lexicographically smallest string among the shortest correct bracket sequences that can be obtained by inserting some number of ( and ) into S.

Sample Input 1

3
())

Sample Output 1

(())

Sample Input 2

6
)))())

Sample Output 2

(((()))())

Sample Input 3

8
))))((((

Sample Output 3

(((())))(((())))

Solution in Ruby (2.3.3) (judge verdict: Accepted)

require 'prime'
require 'set'
require 'tsort'
include Math
def max(a,b);              a > b ? a : b                              end
def min(a,b);              a < b ?  a : b                             end
def swap(a,b);             a, b = b, a                                end
def gif;                   gets.to_i                                  end
def gff;                   gets.to_f                                  end
def gsf;                   gets.chomp                                 end
def gi;                    gets.split.map(&:to_i)                     end
def gf;                    gets.split.map(&:to_f)                     end
def gs;                    gets.chomp.split.map(&:to_s)               end
def gc;                    gets.chomp.split('')                       end
def pr(num);               num.prime_division                         end
def digit(num);            num.to_s.length                            end
def array(s,ini=nil);      Array.new(s){ini}                          end
def darray(s1,s2,ini=nil); Array.new(s1){Array.new(s2){ini}}          end
def rep(num);              num.times{|i|yield(i)}                     end
def repl(st,en,n=1);       st.step(en,n){|i|yield(i)}                 end

st = []
n = gif
str = gc
str.each do |s|
  if s == ')'
    if st.last == '('
      st.pop
    else
      st << s
    end
  else
    st << s
  end
end

st.each do |s|
  if s == '('
    str << ')'
  else
    str.unshift '('
  end
end

puts str.join