Lost Array
Description
A long time ago, Spyofgame invented the famous array $a$ ($1$-indexed) of length $n$ that contains information about the world and life. After that, he decided to convert it into the matrix $b$ ($0$-indexed) of size $(n + 1) \times (n + 1)$ which contains information about the world, life and beyond.
Spyofgame converted $a$ into $b$ with the following rules.
- $b_{i,0} = 0$ if $0 \leq i \leq n$;
- $b_{0,i} = a_{i}$ if $1 \leq i \leq n$;
- $b_{i,j} = b_{i,j-1} \oplus b_{i-1,j}$ if $1 \leq i, j \leq n$.
Here $\oplus$ denotes the bitwise XOR operation.
Today, archaeologists have discovered the famous matrix $b$. However, many elements of the matrix has been lost. They only know the values of $b_{i,n}$ for $1 \leq i \leq n$ (note that these are some elements of the last column, not the last row).
The archaeologists want to know what a possible array of $a$ is. Can you help them reconstruct any array that could be $a$?
The first line contains a single integer $n$ ($1 \leq n \leq 5 \cdot 10^5$).
The second line contains $n$ integers $b_{1,n}, b_{2,n}, \ldots, b_{n,n}$ ($0 \leq b_{i,n} < 2^{30}$).
If some array $a$ is consistent with the information, print a line containing $n$ integers $a_1, a_2, \ldots, a_n$. If there are multiple solutions, output any.
If such an array does not exist, output $-1$ instead.
Input
The first line contains a single integer $n$ ($1 \leq n \leq 5 \cdot 10^5$).
The second line contains $n$ integers $b_{1,n}, b_{2,n}, \ldots, b_{n,n}$ ($0 \leq b_{i,n} < 2^{30}$).
Output
If some array $a$ is consistent with the information, print a line containing $n$ integers $a_1, a_2, \ldots, a_n$. If there are multiple solutions, output any.
If such an array does not exist, output $-1$ instead.
Samples
3
0 2 1
1 2 3
1
199633
199633
10
346484077 532933626 858787727 369947090 299437981 416813461 865836801 141384800 157794568 691345607
725081944 922153789 481174947 427448285 516570428 509717938 855104873 280317429 281091129 1050390365
Note
If we let $a = [1,2,3]$, then $b$ will be:
| $\bf{0}$ | $\bf{1}$ | $\bf{2}$ | $\bf{3}$ |
| $\bf{0}$ | $1$ | $3$ | $0$ |
| $\bf{0}$ | $1$ | $2$ | $2$ |
| $\bf{0}$ | $1$ | $3$ | $1$ |
The values of $b_{1,n}, b_{2,n}, \ldots, b_{n,n}$ generated are $[0,2,1]$ which is consistent with what the archaeologists have discovered.