The Harmonization of XOR
Contest: TypeDB Forces 2023 (Div. 1 + Div. 2, Rated, Prizes!)
Difficulty rating: 2100
Tags: bitmasks, constructive algorithms, greedy, math

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:

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

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 format:
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 format:
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".

Constraints:
Time limit: 1.0 s
Memory limit: 256.0 MB

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:

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

Example input:
7
15 6 7
11 4 5
5 3 2
4 1 4
6 1 7
11 5 5
11 6 5

Example 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

Editorial:
First, we observe that three subsequences can combine into one, so we only need to care about the maximum number of subsequences.
Make subsequences in the form of $$$[a,a \oplus x]$$$ as much as possible, leave $$$[x]$$$ alone if possible, and the rest becomes a subsequence. This would be optimal.
Proof:
Let $$$B$$$ be the highest bit of $$$x$$$, i.e. the $$$B$$$-th bit of $$$x$$$ is on.
Let $$$M$$$ be the number of numbers from $$$1$$$ to $$$n$$$ satisfying the $$$B$$$-th bit is on.
Then the number of subsequences must be smaller than or equal to $$$M$$$, since there's at least one $$$B$$$-th-bit-on number in each subsequence.
These $$$B$$$-th-bit-on numbers XOR $$$x$$$ must be smaller than themselves, so we can always obtain $$$M$$$ subsequences.