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E. Neko and Flashback
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

A permutation of length $$$k$$$ is a sequence of $$$k$$$ integers from $$$1$$$ to $$$k$$$ containing each integer exactly once. For example, the sequence $$$[3, 1, 2]$$$ is a permutation of length $$$3$$$.

When Neko was five, he thought of an array $$$a$$$ of $$$n$$$ positive integers and a permutation $$$p$$$ of length $$$n - 1$$$. Then, he performed the following:

  • Constructed an array $$$b$$$ of length $$$n-1$$$, where $$$b_i = \min(a_i, a_{i+1})$$$.
  • Constructed an array $$$c$$$ of length $$$n-1$$$, where $$$c_i = \max(a_i, a_{i+1})$$$.
  • Constructed an array $$$b'$$$ of length $$$n-1$$$, where $$$b'_i = b_{p_i}$$$.
  • Constructed an array $$$c'$$$ of length $$$n-1$$$, where $$$c'_i = c_{p_i}$$$.

For example, if the array $$$a$$$ was $$$[3, 4, 6, 5, 7]$$$ and permutation $$$p$$$ was $$$[2, 4, 1, 3]$$$, then Neko would have constructed the following arrays:

  • $$$b = [3, 4, 5, 5]$$$
  • $$$c = [4, 6, 6, 7]$$$
  • $$$b' = [4, 5, 3, 5]$$$
  • $$$c' = [6, 7, 4, 6]$$$

Then, he wrote two arrays $$$b'$$$ and $$$c'$$$ on a piece of paper and forgot about it. 14 years later, when he was cleaning up his room, he discovered this old piece of paper with two arrays $$$b'$$$ and $$$c'$$$ written on it. However he can't remember the array $$$a$$$ and permutation $$$p$$$ he used.

In case Neko made a mistake and there is no array $$$a$$$ and permutation $$$p$$$ resulting in such $$$b'$$$ and $$$c'$$$, print -1. Otherwise, help him recover any possible array $$$a$$$.

Input

The first line contains an integer $$$n$$$ ($$$2 \leq n \leq 10^5$$$) — the number of elements in array $$$a$$$.

The second line contains $$$n-1$$$ integers $$$b'_1, b'_2, \ldots, b'_{n-1}$$$ ($$$1 \leq b'_i \leq 10^9$$$).

The third line contains $$$n-1$$$ integers $$$c'_1, c'_2, \ldots, c'_{n-1}$$$ ($$$1 \leq c'_i \leq 10^9$$$).

Output

If Neko made a mistake and there is no array $$$a$$$ and a permutation $$$p$$$ leading to the $$$b'$$$ and $$$c'$$$, print -1. Otherwise, print $$$n$$$ positive integers $$$a_i$$$ ($$$1 \le a_i \le 10^9$$$), denoting the elements of the array $$$a$$$.

If there are multiple possible solutions, print any of them.

Examples
Input
5
4 5 3 5
6 7 4 6
Output
3 4 6 5 7 
Input
3
2 4
3 2
Output
-1
Input
8
2 3 1 1 2 4 3
3 4 4 2 5 5 4
Output
3 4 5 2 1 4 3 2 
Note

The first example is explained is the problem statement.

In the third example, for $$$a = [3, 4, 5, 2, 1, 4, 3, 2]$$$, a possible permutation $$$p$$$ is $$$[7, 1, 5, 4, 3, 2, 6]$$$. In that case, Neko would have constructed the following arrays:

  • $$$b = [3, 4, 2, 1, 1, 3, 2]$$$
  • $$$c = [4, 5, 5, 2, 4, 4, 3]$$$
  • $$$b' = [2, 3, 1, 1, 2, 4, 3]$$$
  • $$$c' = [3, 4, 4, 2, 5, 5, 4]$$$