codeforces 460D：Little Victor and Set

Description

Little Victor adores the sets theory. Let us remind you that a set is a group of numbers where all numbers are pairwise distinct. Today Victor wants to find a set of integers S that has the following properties:

for all x

the following inequality holds l ≤ x ≤ r;

1 ≤ |S| ≤ k;

lets denote the i-th element of the set S as si; value

must be as small as possible.

Help Victor find the described set.

Input
The first line contains three space-separated integers l, r, k (1 ≤ l ≤ r ≤ 1012; 1 ≤ k ≤ min(106, r - l + 1)).

Output
Print the minimum possible value of f(S). Then print the cardinality of set |S|. Then print the elements of the set in any order.

If there are multiple optimal sets, you can print any of them.

Examples

Input

8 15 3

Output

1
2
10 11

Input

8 30 7

Output

0
5
14 9 28 11 16

Note
Operation

represents the operation of bitwise exclusive OR. In other words, it is the XOR operation.

正解：分类讨论

解题报告：

　　今天考试原题，直接上题解：

　　分类讨论。
　　首先注意到(2x) ⊕ (2x + 1) = 1。一个直接推论就是(2x) ⊕ (2x + 1) ⊕ (2x + 2) ⊕(2x + 3) = 0。而k ≥ 5的时候一定可以选出四个数其异或和为0(例如,如果l是偶数,那么选l, l + 1, l + 2, l + 3,否则选l + 1, l + 2, l + 3, l + 4。这样总是合法的,因为r ≥ l + 4)。
　　然后按照k的值分类讨论。
　　k = 1。答案就是l。
　　k = 2。如果r = l + 1,那么答案是min⁡(l ⊕ r, l),其中l表示只选一个数,l ⊕ r表示选两个数。否则答案一定为1。当r > l + 1的时候,如果l是偶数,那么答案是l ⊕ (l + 1),否则答案是(l + 1) ⊕ (l + 2)。
　　k = 3。答案不会超过1,因为从[l, r]中选两个数一定可以使得答案为1。我们关心的是三个数的异或和为0的情况。这3个数的最高位不可能相同(否则异或起来的最高位为1)。设这三个数为x, y, z(x < y <z),且y = 2 ^k + b, z = 2^ k + c, b < c。那么x ⊕ b ⊕ c = 0。为了让l ≤ x, y, z ≤ r,我们需要使得x尽量大而c尽量小。假设x ≥ 2 ^(k−1) ,那么可以推出z ≥ 2^( k−1) + 2^ k 。我们需要使x尽量大而c尽量小,那么令x = 2 ^(k − 1), z = 2 ^(k−1) + 2^ k ,可以得到y = z − 1。满足要求。那么如果x < 2^( k−1) 呢?我们会发现:此时z没必要≥ 2^ k ,因为取y = 2^( k−1 + b), z = 2 ^(k−1 )+ c依然满足条件。
　　所以枚举k并检查x = 2 ^k − 1, z = 2^ k + 2^( k−1) , y = z − 1这三个数是否在[l, r]内,如果在的话,就找到了一组解。
　　k = 4。如果r − l ≥ 4,那么答案一定为0。否则,枚举{x|x ∈ Z ∧ l ≤ x ≤ r}的所有子集,求异或和的最小值。

　　我的代码后面分类讨论部分写丑了。　　

//It is made by jump~
#include <iostream>
#include <cstdlib>
#include <cstring>
#include <cstdio>
#include <cmath>
#include <algorithm>
#include <ctime>
#include <vector>
#include <queue>
#include <map>
#include <set>
using namespace std;
typedef long long LL;
LL l,r,len;
int k;
LL jilu1,jilu2,jilu3,num;

inline int getint()
{
int w=0,q=0;
char c=getchar();
while((c<'0' || c>'9') && c!='-') c=getchar();
if (c=='-')  q=1, c=getchar();
while (c>='0' && c<='9') w=w*10+c-'0', c=getchar();
return q ? -w : w;
}

inline LL getlong()
{
LL w=0,q=0;
char c=getchar();
while((c<'0' || c>'9') && c!='-') c=getchar();
if (c=='-')  q=1, c=getchar();
while (c>='0' && c<='9') w=w*10+c-'0', c=getchar();
return q ? -w : w;
}

inline LL check(){
LL t=0;
while( ((1LL<<t)-1LL) < l) t++;
if(( (1LL<<t)+(1LL<<(t-1LL)) )<=r) return t;
return -1;
}

inline void work(){
LL now;
l=getlong(); r=getlong(); k=getint(); len=r-l+1;
if(k==1) printf("%I64d 1\n%I64d",l,l);
else if(k==2) {
if(l&1) {
if(r-l+1==2){
if((l^r)<l)  printf("%I64d 2\n%I64d %I64d",l^r,l,r);
else printf("%I64d 1\n%I64d",l,l);
}
else  printf("1 2\n%I64d %I64d",l+1,l+2);
}
else printf("1 2\n%I64d %I64d",l,l+1);
}
else if(k==3) {
if((now=check())!=-1) {
printf("0 3\n"); printf("%I64d ",(1LL<<now)-1LL);
printf("%I64d %I64d",(1LL<<now)+(1LL<<(now-1LL))-1LL, (1LL<<now)+(1LL<<(now-1LL) ) );
}
else { printf("1 2\n"); if(l&1) printf("%I64d %I64d",l+1,l+2); else printf("%I64d %I64d",l,l+1);  }
}
else{
if(l&1) {
if(r-l+1>4) printf("0 4\n%I64d %I64d %I64d %I64d",l+1,l+2,l+3,l+4);
else{
now=l; num=1; for(int i=0;i<4;i++) for(int j=i+1;j<4;j++) if( ((l+i)^(l+j)) < now) now=((l+i)^(l+j)),num=2,jilu1=l+i,jilu2=l+j;
if( (l^(l+1)^(l+2) ) < now) now=(l^(l+1)^(l+2) ),num=3,jilu1=l,jilu2=l+1,jilu3=l+2; if( (l^(l+1)^(l+3) ) < now) now=(l^(l+1)^(l+3) ),num=3,jilu1=l,jilu2=l+1,jilu3=l+3;
if( (l^(l+2)^(l+3) ) < now) now=(l^(l+2)^(l+3) ),num=3,jilu1=l,jilu2=l+2,jilu3=l+3; if( ((l+1)^(l+2)^(l+3) ) < now) now=((l+1)^(l+2)^(l+3)),num=3,jilu1=l+1,jilu2=l+2,jilu3=l+3;
if( (l^(l+1)^(l+2)^(l+3) ) < now) now=(l^(l+1)^(l+2)^(l+3) ),num=4;
printf("%I64d %I64d\n",now,num); if(num==1) printf("%I64d",now); else if(num==2) printf("%I64d %I64d",jilu1,jilu2); else if(num==3) printf("%I64d %I64d %I64d",jilu1,jilu2,jilu3); else printf("%I64d %I64d %I64d %I64d",l,l+1,l+2,l+3);
}
}
else printf("0 4\n%I64d %I64d %I64d %I64d",l,l+1,l+2,l+3);
}
}

int main()
{
work();
return 0;
}