Bitwise

Barr the Bear is playing the game *Bits* with Swanky Shen!

Bits is a very simple game. At the start, a circle of $N$ non-negative integers $A_1$, $A_2$, $A_3$, $\ldots $, $A_ N$ is shown to both players. That is, to the left of the integer $A_ i$ is the integer $A_{i-1}$ if $i > 1$; and the integer $A_ N$ otherwise. To the right of the integer $A_ i$ is the integer $A_{i+1}$ if $i < N$; and the integer $A_1$ otherwise. Also an integer $K$ is given to both players.

To win this game, one must divide the circle of integers
into exactly $K$
contiguous non-empty sections, such that the bitwise `AND` of the *powers* of all
sections is maximized. The power of a contiguous section of
integers is the bitwise `OR` of all
integers in that section.

Barr the Bear is lazy and knows that you are wise with bits. Hence, he has hired you to help him to win the game!

**Note:** The binary bitwise operators
`OR` and `AND`
operate on the base-$2$
representation of the integers and correspond to the operators
`|` and `&`
respectively in C++ or Java.

The first line contains integers $N$ and $K$ ($1 \leq K \leq N \leq 5\cdot 10^5$), namely the number of integers and the number of contiguous non-empty sections required.

The next line contains $N$ integers, the $i^\textrm {th}$ of which is the integer $A_ i$ ($0 \leq A_ i \leq 10^9$).

Output a single integer in one line: The maximum bitwise
`AND` of the powers of the sections in
an optimal division of the circle of integers.

In the first sample, the circle is $(2, 3, 4, 1)$. A possible division is
$(3, 4)$ and $(1, 2)$. $(3, 4)$ has power $7$ and $(1, 2)$ has power $3$. The bitwise `AND` of $7$
and $3$ is $3$. Note that a section can possibly
wrap around the circle.

In the second sample, a possible division is $(2, 2, 4)$, $(4)$, $(4, 2)$. The sectionsâ€™ powers are
$6$, $4$ and $6$ respectively, which have bitwise
`AND` of $4$. Note that we require the sections
to be contiguous integers, so the division $(2, 4)$, $(2, 4)$, $(2, 4)$ is not permissible.

In the third sample, we can only have one section. This section will have all the integers, and thus have power $3$.

Sample Input 1 | Sample Output 1 |
---|---|

4 2 2 3 4 1 |
3 |

Sample Input 2 | Sample Output 2 |
---|---|

6 3 2 2 2 4 4 4 |
4 |

Sample Input 3 | Sample Output 3 |
---|---|

4 1 0 1 2 3 |
3 |