You are given a bracket sequence s consisting of n opening '(' and closing ')' brackets.

A regular bracket sequence is a bracket sequence that can be transformed into a correct arithmetic expression by inserting characters '1' and '+' between the original characters of the sequence. For example, bracket sequences "()()", "(())" are regular (the resulting expressions are: "(1)+(1)", "((1+1)+1)"), and ")(" and "(" are not.

You can change the type of some bracket s_i. It means that if s_i =  ')' then you can change it to '(' and vice versa.

Your task is to calculate the number of positions i such that if you change the type of the i-th bracket, then the resulting bracket sequence becomes regular.

Input

The first line of the input contains one integer n (1 ≤ n ≤ 10^6) — the length of the bracket sequence.

The second line of the input contains the string s consisting of n opening '(' and closing ')' brackets.

Output

Print one integer — the number of positions i such that if you change the type of the i-th bracket, then the resulting bracket sequence becomes regular.

Examples

Input


6
(((())


Output


3


Input


6
()()()


Output


0


Input


1
)


Output


0


Input


8
)))(((((


Output


0
inf = 10**6+10


def main():
    n = int(input())
    a = list(input())
    dif = [0]*(n)
    s = 0
    ldif = [inf]*(n+1)
    rdif = [inf]*(n+1)

    for i in range(n):
        if a[i] == "(":
            s += 1
        else:
            s -= 1
        dif[i] = s
        ldif[i] = min(ldif[i-1], s)

    for i in range(n):
        rdif[n-1-i] = min(rdif[n-i], dif[-(i+1)])
    ans = 0
    if abs(dif[-1]) != 2:
        print(0)
        return

    elif dif[-1] == 2:
        #"(" to ")"
        for i in range(n):
            if a[i] == "(" and ldif[i-1] >= 0 and rdif[i] >= 2:
                ans += 1
        print(ans)
        return

    elif dif[-1] == -2:
        # ")" to "("
        for i in range(n):
            if a[i] == ")" and ldif[i-1] >= 0 and rdif[i] >= -2:
                ans += 1
        print(ans)
        return


if __name__ == "__main__":
    main()

