Maximal AND
Description
Let $\mathsf{AND}$ denote the bitwise AND operation, and $\mathsf{OR}$ denote the bitwise OR operation.
You are given an array $a$ of length $n$ and a non-negative integer $k$. You can perform at most $k$ operations on the array of the following type:
- Select an index $i$ ($1 \leq i \leq n$) and replace $a_i$ with $a_i$ $\mathsf{OR}$ $2^j$ where $j$ is any integer between $0$ and $30$ inclusive. In other words, in an operation you can choose an index $i$ ($1 \leq i \leq n$) and set the $j$-th bit of $a_i$ to $1$ ($0 \leq j \leq 30$).
Output the maximum possible value of $a_1$ $\mathsf{AND}$ $a_2$ $\mathsf{AND}$ $\dots$ $\mathsf{AND}$ $a_n$ after performing at most $k$ operations.
The first line of the input contains a single integer $t$ ($1 \le t \le 100$) — the number of test cases. The description of test cases follows.
The first line of each test case contains the integers $n$ and $k$ ($1 \le n \le 2 \cdot 10^5$, $0 \le k \le 10^9$).
Then a single line follows, containing $n$ integers describing the arrays $a$ ($0 \leq a_i < 2^{31}$).
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
For each test case, output a single line containing the maximum possible $\mathsf{AND}$ value of $a_1$ $\mathsf{AND}$ $a_2$ $\mathsf{AND}$ $\dots$ $\mathsf{AND}$ $a_n$ after performing at most $k$ operations.
Input
The first line of the input contains a single integer $t$ ($1 \le t \le 100$) — the number of test cases. The description of test cases follows.
The first line of each test case contains the integers $n$ and $k$ ($1 \le n \le 2 \cdot 10^5$, $0 \le k \le 10^9$).
Then a single line follows, containing $n$ integers describing the arrays $a$ ($0 \leq a_i < 2^{31}$).
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
Output
For each test case, output a single line containing the maximum possible $\mathsf{AND}$ value of $a_1$ $\mathsf{AND}$ $a_2$ $\mathsf{AND}$ $\dots$ $\mathsf{AND}$ $a_n$ after performing at most $k$ operations.
Samples
4
3 2
2 1 1
7 0
4 6 6 28 6 6 12
1 30
0
4 4
3 1 3 1
2
4
2147483646
1073741825
Note
For the first test case, we can set the bit $1$ ($2^1$) of the last $2$ elements using the $2$ operations, thus obtaining the array [$2$, $3$, $3$], which has $\mathsf{AND}$ value equal to $2$.
For the second test case, we can't perform any operations so the answer is just the $\mathsf{AND}$ of the whole array which is $4$.