Machine

The ancient Egyptian computer scientists built several machines that shuffled arrays of integers. Thousands of years later, archaeologists discovered one of them and examined it.

The machine takes an array $A$ of $N$ integers as input. It has a built-in permutation $P$ of the numbers from $0$ to $N-1$, which it uses to shuffle the array. $P$ is an array of length $N$ in which each integer from $0$ to $N-1$ appears exactly once.

Due to corrosion, the machine not only shuffles the numbers but also XORs each one with an unknown number $X$. The machine returns an array $B$ of length $N$ such that

$B[i] = A[P[i]] \oplus X \quad (0 \le i < N)$

Here $\oplus$ is the bitwise XOR operator. For non-negative integers $c$ and $d$, the $j$-th bit of $c \oplus d$ is 1 exactly when the $j$-th bits of $c$ and $d$ differ. $X$ is fixed, so it does not change between uses of the machine.

The archaeologists want to know the built-in permutation $P$. Use the machine to find $P$. Each subtask limits how many times the machine can be used and the largest number you can put in array $A$.

• $1 \le T \le 100$
• $3 \le N \le 128$
• $0 \le X \le 255$
• $0 \le P[i] < N$ for each $0 \le i < N$
• $P[i] \ne P[j]$ for all $0 \le i < j < N$