The Harmonization of XOR

1000ms

262144K

Description:

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:

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

Input:

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 $$$2 \cdot 10^5$$$.

Output:

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".

Sample Input:

Sample Output:

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

Note:

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:

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

Informação

Provedor Codeforces

Código CF1787E