CF1787E.The Harmonization of XOR

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题目描述

You are given an array of exactly $n$ numbers $[1,2,3,\ldots,n]$ along with integers $k$ and $x$ .

Partition the array in exactly $k$ non-empty disjoint subsequences such that the bitwise XOR of all numbers in each subsequence is $x$ , and each number is in exactly one subsequence. Notice that there are no constraints on the length of each subsequence.

A sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements.

For example, for $n = 15$ , $k = 6$ , $x = 7$ , the following scheme is valid:

$[6,10,11]$ , $6 \oplus 10 \oplus 11 = 7$ ,
$[5,12,14]$ , $5 \oplus 12 \oplus 14 = 7$ ,
$[3,9,13]$ , $3 \oplus 9 \oplus 13 = 7$ ,
$[1,2,4]$ , $1 \oplus 2 \oplus 4 = 7$ ,
$[8,15]$ , $8 \oplus 15 = 7$ ,
$[7]$ , $7 = 7$ ,

where $\oplus$ represents the bitwise XOR operation.The following scheme is invalid, since $8$ , $15$ do not appear:

$[6,10,11]$ , $6 \oplus 10 \oplus 11 = 7$ ,
$[5,12,14]$ , $5 \oplus 12 \oplus 14 = 7$ ,
$[3,9,13]$ , $3 \oplus 9 \oplus 13 = 7$ ,
$[1,2,4]$ , $1 \oplus 2 \oplus 4 = 7$ ,
$[7]$ , $7 = 7$ .

The following scheme is invalid, since $3$ appears twice, and $1$ , $2$ do not appear:

$[6,10,11]$ , $6 \oplus 10 \oplus 11 = 7$ ,
$[5,12,14]$ , $5 \oplus 12 \oplus 14 = 7$ ,
$[3,9,13]$ , $3 \oplus 9 \oplus 13 = 7$ ,
$[3,4]$ , $3 \oplus 4 = 7$ ,
$[8,15]$ , $8 \oplus 15 = 7$ ,
$[7]$ , $7 = 7$ .

输入格式

Each test contains multiple test cases. The first line contains an integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases.

The first and the only line of each test case contains three integers $n$ , $k$ , $x$ ( $1 \le k \le n \le 2 \cdot 10^5$ ; $1\le x \le 10^9$ ) — the length of the array, the number of subsequences and the required XOR.

It's guaranteed that the sum of $n$ does not exceed $2 \cdot 10^5$ .

输出格式

For each test case, if it is possible to partition the sequence, print "YES" in the first line. In the $i$ -th of the following $k$ lines first print the length $s_i$ of the $i$ -th subsequence, then print $s_i$ integers, representing the elements in the $i$ -th subsequence. If there are multiple answers, print any. Note that you can print a subsequence in any order.

If it is not possible to partition the sequence, print "NO".

输入输出样例

输入#1

输出#1

YES
3 6 10 11
3 5 12 14
3 3 9 13
3 1 2 4
2 8 15
1 7
YES
2 1 4
2 2 7
2 3 6
5 5 8 9 10 11
NO
YES
4 1 2 3 4
YES
6 1 2 3 4 5 6
NO
NO

说明/提示

In the first test case, we construct the following $6$ subsequences:

$[6,10,11]$ , $6 \oplus 10 \oplus 11 = 7$ ,
$[5,12,14]$ , $5 \oplus 12 \oplus 14 = 7$ ,
$[3,9,13]$ , $3 \oplus 9 \oplus 13 = 7$ ,
$[1,2,4]$ , $1 \oplus 2 \oplus 4 = 7$ ,
$[8,15]$ , $8 \oplus 15 = 7$ ,
$[7]$ , $7 = 7$ .

In the second test case, we construct the following $4$ subsequences:

$[1,4]$ , $1 \oplus 4 = 5$ ,
$[2,7]$ , $2 \oplus 7 = 5$ ,
$[3,6]$ , $3 \oplus 6 = 5$ ,
$[5,8,9,10,11]$ , $5 \oplus 8 \oplus 9 \oplus 10 \oplus 11 = 5$ .

The following solution is considered correct in this test case as well:

$[1,4]$ , $1 \oplus 4 = 5$ ,
$[2,7]$ , $2 \oplus 7 = 5$ ,
$[5]$ , $5 = 5$ ,
$[3,6,8,9,10,11]$ , $3 \oplus 6 \oplus 8 \oplus 9 \oplus 10 \oplus 11 = 5$ .

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