description: A bracket sequence is a string containing only characters "(" and ")". A regular bracket sequence (or, shortly, an RBS) 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 "()()" and "(())" are regular (the resulting expressions are: "(1)+(1)" and "((1+1)+1)"); 
    * bracket sequences ")(", "(" and ")" are not. 
  
  
  
  Let's denote the concatenation of two strings x and y as x+y. For example, "()()" + ")(" = "()())(".
  
  You are given n bracket sequences s_1, s_2, ..., s_n. You can rearrange them in any order (you can rearrange only the strings themselves, but not the characters in them).
  
  Your task is to rearrange the strings in such a way that the string s_1 + s_2 + ... + s_n has as many non-empty prefixes that are RBS as possible.
  
  Input
  
  The first line contains a single integer n (1 ≤ n ≤ 20).
  
  Then n lines follow, the i-th of them contains s_i — a bracket sequence (a string consisting of characters "(" and/or ")". All sequences s_i are non-empty, their total length does not exceed 4 ⋅ 10^5.
  
  Output
  
  Print one integer — the maximum number of non-empty prefixes that are RBS for the string s_1 + s_2 + ... + s_n, if the strings s_1, s_2, ..., s_n can be rearranged arbitrarily.
  
  Examples
  
  Input
  
  
  2
  (
  )
  
  
  Output
  
  
  1
  
  
  Input
  
  
  4
  ()()())
  (
  (
  )
  
  
  Output
  
  
  4
  
  
  Input
  
  
  1
  (())
  
  
  Output
  
  
  1
  
  
  Input
  
  
  1
  )(()
  
  
  Output
  
  
  0
  
  Note
  
  In the first example, you can concatenate the strings as follows: "(" + ")" = "()", the resulting string will have one prefix, that is an RBS: "()".
  
  In the second example, you can concatenate the strings as follows: "(" + ")" + "()()())" + "(" = "()()()())(", the resulting string will have four prefixes that are RBS: "()", "()()", "()()()", "()()()()".
  
  The third and the fourth examples contain only one string each, so the order is fixed.
solution: import sys
  from sys import stdin
  
  n = int(stdin.readline())
  
  s = [list(stdin.readline()[:-1]) for i in range(n)]
  
  smin = [0] * n
  smnum = [0] * n
  h = [0] * n
  
  cnts = [{} for i in range(n)]
  
  for i in range(n):
  
      ts = s[i]
  
      nmin = 0
  
      nh = 0
      for c in ts:
          if c == "(":
              nh += 1
          else:
              nh -= 1
  
          if smin[i] > nh:
              smin[i] = nh
              smnum[i] = 0
          if smin[i] == nh:
              smnum[i] += 1
  
          if nh == smin[i]:
              if nh not in cnts[i]:
                  cnts[i][nh] = 0
              cnts[i][nh] += 1
  
      h[i] = nh
  
  dp = [float("-inf")] * (2**n)
  dp[0] = 0
  
  ans = 0
  
  for i in range(2**n):
  
      nh = 0
      for j in range(n):
          if 2**j & i > 0:
              nh += h[j]
  
      for j in range(n):
          if 2**j & i == 0 and nh + smin[j] >= 0:
              if nh + smin[j] == 0:
                  dp[2**j | i] = max(dp[2**j | i] , dp[i] + smnum[j])
              else:
                  dp[2**j | i] = max(dp[2**j | i] , dp[i])
          elif 2**j & i == 0 and -1*nh in cnts[j]:
              ans = max(ans , dp[i] + cnts[j][-1*nh])
  
  #print (dp)
  ans = max(ans , max(dp))
  print (ans)
  