Background

Xiaofang always has many doubts.

Problem Description

On Little C's birthday a few years ago, he received a gift: a string of length $n$ consisting only of $\text{(}$ and $\text{)}$, in which exactly $k$ substrings are valid bracket sequences.

Since Xiaofang is very curious about what this gift looks like, you need to construct such a string, or output $-1$ to indicate that there is no solution.

A valid bracket sequence satisfies the following conditions:

• $\text{()}$ is a valid bracket sequence.
• If $\text{A}$ and $\text{B}$ are valid bracket sequences, then $\text{(A)}$ and $\text{AB}$ are valid bracket sequences.
• All other sequences are not valid bracket sequences.

Xiaofang has many doubts, so he will ask you $T$ times.

Input Format

The first line contains an integer $T$ representing the number of queries.

The next $T$ lines each contain two integers $n$ and $k$ ($1\le n,k \le 10^5$, $\sum n\le 10^6$).

The input and output size of this problem is large, so please use fast I/O.

Output Format

For each query, if a solution exists, output a string of length $n$ on one line; otherwise output $-1$.

3
5 2
5 3
5 4

(())(
()()(
-1

Hint

Translated by ChatGPT 5