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Minimum Add to Make Parentheses Valid
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Given a string S of '(' and ')' parentheses, we add the minimum number of parentheses ( '(' or ')', and in any positions ) so that the resulting parentheses string is valid.

Formally, a parentheses string is valid if and only if:

  • It is the empty string, or
  • It can be written as AB (A concatenated with B), where A and B are valid strings, or
  • It can be written as (A), where A is a valid string.

Given a parentheses string, return the minimum number of parentheses we must add to make the resulting string valid.

SAMPLE INPUT
()))((
SAMPLE OUTPUT
4
Explanation

The given string can be balanced to "((()))(())", requiring 4 more paranthesis minimum.

Time Limit: 5.0 sec(s) for each input file.
Memory Limit: 256 MB
Source Limit: 1024 KB

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