A15733 | The Harmonization of XOR

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](https://en.wikipedia.org/wiki/Bitwise_operation#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:

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

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 $2 \cdot 10^5$ .

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".