Merge Elements

You are given an array $A=[A_{1}, \dots, A_{N}]$ of $N$ non-negative integers. Perform the following operation $K$ times on $A$.

• Choose two adjacent elements that remain in the array, apply bitwise $\textrm{OR}$ to them, and replace the two elements with the result. In other words, if the current length of the array is $L$, choose an integer $i(1 \leq i < L)$ and change $[A_1, \dots, A_i, A_{i+1}, \dots, A_{L}]$ to $[A_1, \dots, A_i \mid A_{i+1}, \dots, A_{L}]$. Each operation reduces the length of the array by $1$.

After $K$ operations, find the maximum possible value of the bitwise $\textrm{AND}$ of all $N-K$ remaining elements.

The first line contains $N$ and $K$, separated by a space. ($2 \leq N \leq 500\,000$; $1 \leq K \leq N-1$)

The second line contains the elements $A_1, A_2, \dots, A_{N}$, separated by spaces. ($0 \leq A_{i} \leq 10^{9}$)

Print the maximum value of the bitwise $\textrm{AND}$ of the $N-K$ elements that remain after $K$ operations.

Bitwise operators work on each bit separately.

• The bitwise $\textrm{AND}$ operation ($\&$) works on each bit of two numbers. The result bit is $1$ only when both bits are $1$, and $0$ otherwise.

  • For example, $\begin{array}{rcl} 0101_2 & = & 5 \\ \\& \ 0011_2 & = & 3 \\ \hline 0001_2 & = & 1 \end{array}$

• The bitwise $\textrm{OR}$ operation ($|$) works on each bit of two numbers. The result bit is $0$ only when both bits are $0$, and $1$ otherwise.

  • For example, $\begin{array}{rcl} 0110_2 & = & 6 \\ \ | \ 1100_2 & = & 12 \\ \hline 1110_2 & = & 14 \end{array}$