CF1787E 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".

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 $ .