XOR Palindromes
Description
You are given a binary string $s$ of length $n$ (a string that consists only of $0$ and $1$). A number $x$ is good if there exists a binary string $l$ of length $n$, containing $x$ ones, such that if each symbol $s_i$ is replaced by $s_i \oplus l_i$ (where $\oplus$ denotes the bitwise XOR operation), then the string $s$ becomes a palindrome.
You need to output a binary string $t$ of length $n+1$, where $t_i$ ($0 \leq i \leq n$) is equal to $1$ if number $i$ is good, and $0$ otherwise.
A palindrome is a string that reads the same from left to right as from right to left. For example, 01010, 1111, 0110 are palindromes.
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^5$). The description of the test cases follows.
The first line of each test case contains a single integer $n$ ($1 \le n \le 10^5$).
The second line of each test case contains a binary string $s$ of length $n$.
It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$.
For each test case, output a single line containing a binary string $t$ of length $n+1$ - the answer to the problem.
Input
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^5$). The description of the test cases follows.
The first line of each test case contains a single integer $n$ ($1 \le n \le 10^5$).
The second line of each test case contains a binary string $s$ of length $n$.
It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$.
Output
For each test case, output a single line containing a binary string $t$ of length $n+1$ - the answer to the problem.
5
6
101011
5
00000
9
100100011
3
100
1
1
0010100
111111
0011111100
0110
11
Note
Consider the first example.
- $t_2 = 1$ because we can choose $l = $ 010100, then the string $s$ becomes 111111, which is a palindrome.
- $t_4 = 1$ because we can choose $l = $ 101011.
- It can be shown that for all other $i$, there is no answer, so the remaining symbols are $0$.