P9696 [GDCPC 2023] Swapping Operation

Given a non-negative integer sequence $A = a_1, a_2, \dots, a_n$ of length $n$, define $$ F(A)=\max\limits_{1\leq k

There are multiple test cases. The first line of the input contains an integer $T$ indicating the number of test cases. For each test case: The first line contains an integer $n$ ($2\leq n\leq 10^5$) indicating the length of sequence $A$. The second line contains $n$ integers $a_1, a_2, \cdots, a_n$ ($0\leq a_i\leq 10^9$) indicating the given sequence $A$. It's guaranteed that the sum of $n$ of all test cases will not exceed $10^5$.

For each test case output one line containing one integer indicating the maximum possible value of $F(A)$ after performing at most one swapping operation.

For the first sample test case, we can swap $a_4$ and $a_6$ so the sequence becomes $\{6, 5, 4, 6, 5, 3\}$. We can then choose $k = 5$ so that $F(A) = (6 \,\&\, 5 \,\&\, 4 \,\&\, 6 \,\&\, 5) + (3) = 7$. For the second sample test case, we can swap $a_2$ and $a_4$ so the sequence becomes $\{1, 1, 1, 2, 2, 2\}$. We can then choose $k = 3$ so that $F(A) = (1 \,\&\, 1 \,\&\, 1) + (2 \,\&\, 2 \,\&\, 2) = 3$. For the third sample test case we do not perform the swapping operation. We can then choose $k = 2$ so that $F(A) = (1 \,\&\, 1) + (2 \,\&\, 2 \,\&\, 2) = 3$.