A13513. Maximum Subsequence Value

Ridhiman challenged Ashish to find the maximum valued subsequence of an array $a$ of size $n$ consisting of positive integers.

The value of a non-empty subsequence of $k$ elements of $a$ is defined as $\sum 2^i$ over all integers $i \ge 0$ such that at least $\max(1, k - 2)$ elements of the subsequence have the $i$ -th bit set in their binary representation (value $x$ has the $i$ -th bit set in its binary representation if $\lfloor \frac{x}{2^i} \rfloor \mod 2$ is equal to $1$ ).

Recall that $b$ is a subsequence of $a$ , if $b$ can be obtained by deleting some(possibly zero) elements from $a$ .

Help Ashish find the maximum value he can get by choosing some subsequence of $a$ .

The first line of the input consists of a single integer $n$ $(1 \le n \le 500)$ — the size of $a$ .

The next line consists of $n$ space-separated integers — the elements of the array $(1 \le a_i \le 10^{18})$ .

Print a single integer — the maximum value Ashish can get by choosing some subsequence of $a$ .

For the first test case, Ashish can pick the subsequence $\{{2, 3}\}$ of size $2$ . The binary representation of $2$ is 10 and that of $3$ is 11. Since $\max(k - 2, 1)$ is equal to $1$ , the value of the subsequence is $2^0 + 2^1$ (both $2$ and $3$ have $1$ -st bit set in their binary representation and $3$ has $0$ -th bit set in its binary representation). Note that he could also pick the subsequence $\{{3\}}$ or $\{{2, 1, 3\}}$ .

For the second test case, Ashish can pick the subsequence $\{{3, 4\}}$ with value $7$ .

For the third test case, Ashish can pick the subsequence $\{{1\}}$ with value $1$ .

For the fourth test case, Ashish can pick the subsequence $\{{7, 7\}}$ with value $7$ .