P16742 [GKS 2019 #G] The Equation

The laws of the universe can be represented by an array of $N$ non-negative integers. The i-th of these integers is $A_i$. The universe is *good* if there is a non-negative integer $k$ such that the following equation is satisfied: $(A_1 \text{ xor } k) + (A_2 \text{ xor } k) + ... (A_N \text{ xor } k) \le M$, where xor denotes the bitwise exclusive or. What is the largest value of $k$ for which the universe is good?

The first line of the input gives the number of test cases, $T$. $T$ test cases follow. Each test case begins with a line containing the two integers $N$ and $M$, the number of integers in $A$ and the limit on the equation, respectively. The second line contains $N$ integers, the i-th of which is $A_i$, the i-th integer in the array.

For each test case, output one line containing Case #x: y, where x is the test case number (starting from 1) and y is the largest value of k for which the universe is good, or $-1$ if there is no such $k$.

In sample case #1, the array contains $N = 3$ integers and $M = 27$. The largest possible value of $k$ that gives a good universe is $12$ ($(8 \text{ xor } 12) + (2 \text{ xor } 12) + (4 \text{ xor } 12) = 26$). In sample case #2, the array contains $N = 4$ integers and $M = 45$. The largest possible value of $k$ that gives a good universe is $14$ ($(30 \text{ xor } 14) + (0 \text{ xor } 14) + (4 \text{ xor } 14) + (11 \text{ xor } 14) = 45$). In sample case #3, the array contains $N = 1$ integer and $M = 0$. The largest possible value of $k$ that gives a good universe is $100$ ($100 \text{ xor } 100 = 0$). In sample case #4, there is no value of $k$ that gives a good universe, so the answer is $-1$. ### Limits $1 \le T \le 100$. $1 \le N \le 1000$. **Test set 1 (Visible)** $0 \le M \le 100$. $0 \le A_i \le 100$, for all $i$. **Test set 2 (Hidden)** $0 \le M \le 10^{15}$. $0 \le A_i \le 10^{15}$, for all $i$.