CF1789D.Serval and Shift-Shift-Shift

Serval has two $n$-bit binary integer numbers $a$ and $b$. He wants to share those numbers with Toxel.

Since Toxel likes the number $b$ more, Serval decides to change $a$ into $b$ by some (possibly zero) operations. In an operation, Serval can choose any positive integer $k$ between $1$ and $n$, and change $a$ into one of the following number:

• $a\oplus(a\ll k)$
• $a\oplus(a\gg k)$

In other words, the operation moves every bit of $a$ left or right by $k$ positions, where the overflowed bits are removed, and the missing bits are padded with $0$. The bitwise XOR of the shift result and the original $a$ is assigned back to $a$.

Serval does not have much time. He wants to perform no more than $n$ operations to change $a$ into $b$. Please help him to find out an operation sequence, or determine that it is impossible to change $a$ into $b$ in at most $n$ operations. You do not need to minimize the number of operations.

In this problem, $x\oplus y$ denotes the bitwise XOR operation of $x$ and $y$. $a\ll k$ and $a\gg k$ denote the logical left shift and logical right shift.

Serval 有两个 $n$ 位二进制整数 $a$ 和 $b$。他希望将这两个数分享给 Toxel。

由于 Toxel 更喜欢数字 $b$，Serval 决定通过若干次（可能为零次）操作将 $a$ 变为 $b$。每次操作中，Serval 可以任选一个介于 $1$ 到 $n$ 之间的正整数 $k$，并将 $a$ 替换为以下两个数之一：

• $a\oplus(a\ll k)$
• $a\oplus(a\gg k)$

换言之，该操作将 $a$ 的每一位向左或向右移动 $k$ 个位置，移出边界的位被丢弃，空缺位置补 $0$；然后将移位结果与原 $a$ 进行按位异或，并将结果赋回给 $a$。

Serval 时间有限，他希望至多执行 $n$ 次操作便将 $a$ 变为 $b$。请帮助他找出一个满足要求的操作序列，或判定在至多 $n$ 次操作内无法将 $a$ 变为 $b$。你无需最小化操作次数。

本题中，$x\oplus y$ 表示 $x$ 与 $y$ 的按位异或运算；$a\ll k$ 和 $a\gg k$ 分别表示逻辑左移和逻辑右移。

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1\le t\le2\cdot10^{3}$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($1\le n\le2\cdot10^{3}$) — the number of bits in numbers $a$ and $b$.

The second and the third line of each test case contain a binary string of length $n$, representing $a$ and $b$, respectively. The strings contain only characters 0 and 1.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2\cdot10^{3}$.

每个测试包含多个测试用例。第一行包含测试用例的数量 $t$（$1\le t\le2\cdot10^{3}$）。随后是各测试用例的描述。

每个测试用例的第一行包含一个整数 $n$（$1\le n\le2\cdot10^{3}$）—— 表示数字 $a$ 和 $b$ 的二进制位数。

每个测试用例的第二行和第三行分别包含一个长度为 $n$ 的二进制字符串，表示 $a$ 和 $b$。这些字符串仅由字符 0 和 1 组成。

保证所有测试用例的 $n$ 之和不超过 $2\cdot10^{3}$。

For each test case, if it is impossible to change $a$ into $b$ in at most $n$ operations, print a single integer $-1$.

Otherwise, in the first line, print the number of operations $m$ ($0\le m\le n$).

If $m \gt 0$, in the second line, print $m$ integers $k_{1},k_{2},\dots,k_{m}$ representing the operations. If $1\le k_{i}\le n$, it means logical left shift $a$ by $k_{i}$ positions. If $-n\le k_{i}\le-1$, it means logical right shift $a$ by $-k_{i}$ positions.

If there are multiple solutions, print any of them.

对于每个测试用例，如果无法在至多 $n$ 次操作内将 $a$ 变为 $b$，则输出单个整数 $-1$。

否则，在第一行输出操作次数 $m$（$0\le m\le n$）。

若 $m \gt 0$，则在第二行输出 $m$ 个整数 $k_{1},k_{2},\dots,k_{m}$，表示所执行的操作：若 $1\le k_{i}\le n$，表示对 $a$ 进行逻辑左移 $k_{i}$ 位；若 $-n\le k_{i}\le-1$，表示对 $a$ 进行逻辑右移 $-k_{i}$ 位。

若存在多种解法，输出任意一种即可。

In the first test case:

The first operation changes $a$ into \require{cancel}00111\oplus\cancel{001}11\underline{000}=11111.

The second operation changes $a$ into \require{cancel}11111\oplus\underline{00}111\cancel{11}=11000.

The bits with strikethroughs are overflowed bits that are removed. The bits with underline are padded bits.

In the second test case, $a$ is already equal to $b$, so no operations are needed.

In the third test case, it can be shown that $a$ cannot be changed into $b$.

在第一个测试用例中：

第一次操作将 $a$ 变为 \require{cancel}00111\oplus\cancel{001}11\underline{000}=11111。

第二次操作将 $a$ 变为 \require{cancel}11111\oplus\underline{00}111\cancel{11}=11000。

带删除线的位是溢出而被移除的位；带下划线的位是补零填充的位。

在第二个测试用例中，$a$ 已经等于 $b$，因此无需任何操作。

在第三个测试用例中，可以证明 $a$ 无法变为 $b$。