F. RBStime limit per test3 secondsmemory limit per test512 megabytesinputstandard inputoutputstandard outputA 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, \dots, 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 + \dots + s_n has as many non-empty prefixes that are RBS as possible.InputThe first line contains a single integer n (1 \le n \le 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 \cdot 10^5.OutputPrint one integer — the maximum number of non-empty prefixes that are RBS for the string s_1 + s_2 + \dots + s_n, if the strings s_1, s_2, \dots, s_n can be rearranged arbitrarily.ExamplesInput
2
(
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Output
1
Input
4
()()())
(
(
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Output
4
Input
1
(())
Output
1
Input
1
)(()
Output
0
NoteIn 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.
2
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1 
['binary search', 'bitmasks', 'brute force', 'data structures', 'dp', '*2400']