Problem: This is yet another problem dealing with regular bracket sequences.

We should remind you that a bracket sequence is called regular, if by inserting «+» and «1» into it we can get a correct mathematical expression. For example, sequences «(())()», «()» and «(()(()))» are regular, while «)(», «(()» and «(()))(» are not. 

You are given a string of «(» and «)» characters. You are to find its longest substring that is a regular bracket sequence. You are to find the number of such substrings as well.

Input

The first line of the input file contains a non-empty string, consisting of «(» and «)» characters. Its length does not exceed 106.

Output

Print the length of the longest substring that is a regular bracket sequence, and the number of such substrings. If there are no such substrings, write the only line containing "0 1".

Examples

Input

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


Output

6 2


Input

))(


Output

0 1

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
using namespace std;
const int Maxn = 1000 * 1000 + 5;
int n, arr[Maxn], Min[Maxn];
vector<int> vec, ind[2 * Maxn];
string s;
char c[Maxn];
int main() {
  scanf("%s", &c);
  s = c;
  n = (int)s.size();
  for (int i = 0; i < n; i++) {
    arr[i + 1] = arr[i];
    if (s[i] == ')')
      arr[i + 1]--;
    else
      arr[i + 1]++;
    ind[arr[i + 1] + 1000 * 1000].push_back(i + 1);
  }
  vec.push_back(n);
  for (int i = n - 1; i >= 0; i--) {
    while ((int)vec.size() > 0 && arr[vec.back()] >= arr[i]) vec.pop_back();
    if ((int)vec.size() == 0) {
      if ((int)ind[arr[i] + 1000 * 1000].size() > 0 &&
          i < ind[arr[i] + 1000 * 1000].back())
        Min[i] = ind[arr[i] + 1000 * 1000].back();
      else
        Min[i] = -1;
    } else
      Min[i] = vec.back() - 1;
    if (s[i] == ')') Min[i] = -1;
    vec.push_back(i);
  }
  int Max = 0, Maxnum = 0;
  for (int i = 0; i < n; i++) {
    if (Min[i] != -1) {
      if (Min[i] - i > Max) {
        Max = Min[i] - i;
        Maxnum = 1;
      } else {
        if (Min[i] - i == Max) Maxnum++;
      }
    }
  }
  if (Max == 0) Maxnum = 1;
  printf("%d %d\n", Max, Maxnum);
  return 0;
}