시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
5 초 512 MB 4 4 4 100.000%

## 문제

An n parentheses sequence consists of n "("s and n ")"s.

A valid parentheses sequence is defined as the following:

You can find a way to repeat erasing adjacent pair of parentheses "()" until it becomes empty.

For example, "(())" is a valid parentheses, you can erase the pair on the 2nd and 3rd position and it becomes "()", then you can make it empty.

")()(" is not a valid parentheses, after you erase the pair on the 2nd and 3rd position, it becomes ")(" and you cannot erase any more.

Now, we have all valid n parentheses sequences. Find the k-th smallest sequence in lexicographical order.

For example, here are all valid 3 parentheses sequences in lexicographical order:

((()))

(()())

(())()

()(())

()()()

## 입력

The first line of the input gives the number of test cases, T. T lines follow. Each line represents a test case consisting of 2 integers, n and k.

Limits

• 1 ≤ T ≤ 100.
• 1 ≤ n ≤ 10.
• 1 ≤ k ≤ 100000.

## 출력

For each test case, output one line containing "Case #x: y", where x is the test case number (starting from 1) and y is the k-th smallest parentheses sequence in all valid n parentheses sequences. Output "Doesn't Exist!" when there are less than k different n parentheses sequences.

## 예제 입력 1

3
2 2
3 4
3 6


## 예제 출력 1

Case #1: ()()
Case #2: ()(())
Case #3: Doesn't Exist!