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 Fortran (GNU Fortran 9.2.1) (judge verdict: Wrong Answer)

program prob4
    implicit none
    integer::i, j, l, r, n, mode
    character(len=100)::S
    character(len=300)::ans, tmp
    read(*,*) n
    read(*,*) S
    ans = ""
    l = 0
    r = 0
    do i = 1, n
        if(S(i:i) .eq. "(") then
            if(r > 0) then
                do j = 1, r
                    ans = ")" // trim(ans)
                end do
                do j = 1, r
                    ans = "(" // trim(ans)
                end do
                r = 0
                l = 0
            end if
            l = l + 1
        else
            if(l > 0) then
                do j = 1, l
                    ans = trim(ans) // "("
                end do
                do j = 1, l
                    ans = trim(ans) // ")"
                end do
                l = 0
                r = 0
            end if
            r = r + 1
        end if
    end do
    if(r > 0) then
        do i = 1, r-1
            ans = trim(ans) // ")"
        end do
        do i = 1, r-1
            ans = "(" // trim(ans)
        end do
    else if(l > 0) then
        do i = 1, l-1
            ans = trim(ans) // "("
        end do
        do i = 1, l-1
            ans = trim(ans) // "9"
        end do
    end if
    write(*,'(a)') trim(ans)
    stop
end program