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 Lua (5.3.2) (judge verdict: Accepted)

local n = io.read("*n", "*l")
local s = io.read()
local t = {}
for i = 1, n do t[i] = s:sub(i, i) == "(" and 1 or - 1 end
local mmi = math.min
while true do
  local minval = 0
  local curval = 0
  for i = 1, n do
    curval = curval + t[i]
    minval = mmi(minval, curval)
  end
  if minval < 0 then
    for i = 1, -minval do
      table.insert(t, 1, 1)
    end
    n = n - minval
  elseif 0 < curval then
    for i = 1, curval do
      table.insert(t, -1)
    end
    n = n + curval
  else
    break
  end
end
for i = 1, n do
  t[i] = t[i] == 1 and "(" or ")"
end
print(table.concat(t))