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:
The following scheme is invalid, since $$$8$$$, $$$15$$$ do not appear:
The following scheme is invalid, since $$$3$$$ appears twice, and $$$1$$$, $$$2$$$ do not appear:
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".
715 6 711 4 55 3 24 1 46 1 711 5 511 6 5
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
In the first test case, we construct the following $$$6$$$ subsequences:
In the second test case, we construct the following $$$4$$$ subsequences:
The following solution is considered correct in this test case as well:
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