CF1669H.Maximal AND

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.

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 $4$ .