1095_E. Almost Regular Bracket Sequence
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
```python
n = int(input())
lis = list(input())
if n%2:
    print(0)
else:
    st=[]
    for i in range(n):
        if lis[i]==')' and len(st)>0 and st[-1]=='(':
            st.pop()
        else:
            st.append(lis[i])
    if len(st)==2 and st[0]==st[1]:
        if st[0]=='(':
            pre=[0]*(n+2)
            ans=0
            for i in range(n):
                if lis[i]=='(':
                    pre[i]=pre[i-1]+1
                else:
                    pre[i]=pre[i-1]-1
            c=0
            for i in range(n-1,-1,-1):
                if pre[i]<=1:
                    c=i
                    break
            for i in range(c+1,n,1):
                if pre[i]>pre[i-1]:
                    ans+=1
            print(ans)
        else:
            suf=[0]*(n+10)
            ans=0
#            print(st)
            for i in range(n-1,-1,-1):
                if lis[i]==')':
                    suf[i]=suf[i+1]+1
                else:
                    suf[i]=suf[i+1]-1
            c=0        
            for i in range(n):
                if suf[i]<=1:
                    c=i
                    break
            for i in range(c):
                if suf[i]>suf[i+1]:
                    ans+=1
            print(ans)            
    else:
        print(0)                        
    
 
```