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[ABC117_D] Problem Statement

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Score : $400$ points

Problem Statement

You are given $N$ non-negative integers $A_1, A_2, ..., A_N$ and another non-negative integer $K$ .

For a integer $X$ between $0$ and $K$ (inclusive), let $f(X) = (X$ XOR $A_1)$ $+$ $(X$ XOR $A_2)$ $+$ $...$ $+$ $(X$ XOR $A_N)$ .

Here, for non-negative integers $a$ and $b$ , $a$ XOR $b$ denotes the bitwise exclusive OR of $a$ and $b$ .

Find the maximum value of $f$ .

What is XOR?

The bitwise exclusive OR of $a$ and $b$ , $X$ , is defined as follows:

• When $X$ is written in base two, the digit in the $2^k$ 's place ( $k \geq 0$ ) is $1$ if, when written in base two, exactly one of $A$ and $B$ has $1$ in the $2^k$ 's place, and $0$ otherwise.

For example, $3$ XOR $5 = 6$ . (When written in base two: $011$ XOR $101 = 110$ .)

Constraints

• All values in input are integers.
• $1 \leq N \leq 10^5$
• $0 \leq K \leq 10^{12}$
• $0 \leq A_i \leq 10^{12}$

Input

Input is given from Standard Input in the following format:

$N$   $K$ 

 $A_1$   $A_2$   $...$   $A_N$

Output

Print the maximum value of $f$ .

Sample Input 1

3 7

1 6 3

Sample Output 1

14

The maximum value is: $f(4) = (4$ XOR $1) + (4$ XOR $6) + (4$ XOR $3) = 5 + 2 + 7 = 14$ .

Sample Input 2

4 9

7 4 0 3

Sample Output 2

46

Sample Input 3

1 0

1000000000000

Sample Output 3

1000000000000