Problem Statement

A string of length $2N, S=S_1S_2\dots S_{2N}$, is said to be a parenthesis sequence when S is composed of N (s and N )s.
Additionally, a parenthesis sequence S is said to be correct when it is one of the following.

• An empty string.
• The concatenation of (, A, ) in this order, where A is a correct parenthesis sequence.
• The concatenation of A, B in this order, where A and B are non-empty correct parenthesis sequences.

You are given two permutations P=(P_1,\ P_2,\ $\dots$,\ P_{2N}) and Q=(Q_1,\ Q_2,\ $\dots$,\ Q_{2N}) of the integers from 1 to 2N.
Determine whether there exists $a parenthesis sequence S=S_1S_2\dots S_{2N}$ that satisfies the following conditions.

• P is the lexicographically smallest permutation p of the integers from 1 to 2N such that $S_{p_1}S_{p_2}\dots S_{p_{2N}}$ is a correct parenthesis sequence.
• Q is the lexicographically largest permutation p of the integers from 1 to 2N such that $S_{p_1}S_{p_2}\dots S_{p_{2N}}$ is a correct parenthesis sequence.

If such a parenthesis sequence exists, find one.

Input

Input is given from Standard Input in the following format:
N
$P_1 P_2 \dots P_{2N}$
$Q_1 Q_2 \dots Q_{2N}$

Output

If there exists a parenthesis sequence S that satisfies the conditions above, print one such sequence. If there are multiple such sequences, you may print any of them.
If there is no such sequence, print -1.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq P_i,Q_i \leq 2N$
• Each of P and Q is a permutation of 1 to 2N.
• All values in input are integers.

Sample Input 1

2
1 2 4 3
4 3 1 2

Sample Output 1

())(

For S= ())(, some of the permutations p such that S_{p_1}S_{p_2}S_{p_3}S_{p_4} is a correct parenthesis sequence are p=(1,\ 4,\ 2,\ 3),\ (1,\ 4,\ 3,\ 2). The lexicographically smallest and largest among them are p=(1,\ 2,\ 4,\ 3) and p=(4,\ 3,\ 1,\ 2), satisfying the conditions.

Sample Input 2

2
1 3 2 4
4 3 2 1

Sample Output 2

-1

No S satisfies the conditions.

Sample Input 3

3
2 1 5 3 4 6
6 5 3 4 2 1

Sample Output 3

-1

见题面 Input 部分

见题面 Output 部分