D. Invertible Bracket Sequences
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

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 "()()" and "(())" are regular (the resulting expressions are: "(1)+(1)" and "((1+1)+1)");
  • bracket sequences ")(", "(" and ")" are not.

Let's define the inverse of the bracket sequence as follows: replace all brackets '(' with ')', and vice versa (all brackets ')' with '('). For example, strings "()((" and ")())" are inverses of each other.

You are given a regular bracket sequence $$$s$$$. Calculate the number of pairs of integers $$$(l,r)$$$ ($$$1 \le l \le r \le |s|$$$) such that if you replace the substring of $$$s$$$ from the $$$l$$$-th character to the $$$r$$$-th character (inclusive) with its inverse, $$$s$$$ will still be a regular bracket sequence.

Input

The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases.

The only line of each test case contains a non-empty regular bracket sequence; it consists only of characters '(' and/or ')'.

Additional constraint on the input: the total length of the regular bracket sequences over all test cases doesn't exceed $$$2 \cdot 10^5$$$.

Output

For each test case, print a single integer — the number of pairs $$$(l,r)$$$ meeting the conditions from the statement.

Example
Input
4
(())
()
()()()
(()())(())
Output
1
0
3
13
Note

In the first example, there is only one pair:

  • $$$(2, 3)$$$: (()) $$$\rightarrow$$$ ()().

In the second example, there are no pairs.

In the third example, there are three pairs:

  • $$$(2, 3)$$$: ()()() $$$\rightarrow$$$ (())();
  • $$$(4, 5)$$$: ()()() $$$\rightarrow$$$ ()(());
  • $$$(2, 5)$$$: ()()() $$$\rightarrow$$$ (()());