codeforces#P1789D. Serval and Shift-Shift-Shift
Serval and Shift-Shift-Shift
Description
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.
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 $2\cdot10^{3}$.
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 $m>0$, 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.
Input
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 $2\cdot10^{3}$.
Output
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 $m>0$, 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.
3
5
00111
11000
1
1
1
3
001
000
2
3 -2
0
-1
Note
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 $b$.