CF1713F.Lost Array

My orzlers, we can optimize this problem from $O(S^3)$ to $O\left(T^\frac{5}{9}\right)$ !

— Spyofgame, founder of Orzlim religion

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.

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.