CF1928E Modular Sequence

You are given two integers $ x $ and $ y $ . A sequence $ a $ of length $ n $ is called modular if $ a_1=x $ , and for all $ 1 < i \le n $ the value of $ a_{i} $ is either $ a_{i-1} + y $ or $ a_{i-1} \bmod y $ . Here $ x \bmod y $ denotes the remainder from dividing $ x $ by $ y $ . Determine if there exists a modular sequence of length $ n $ with the sum of its elements equal to $ S $ , and if it exists, find any such sequence.

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 2 \cdot 10^4 $ ). The description of the test cases follows. The first and only line of each test case contains four integers $ n $ , $ x $ , $ y $ , and $ s $ ( $ 1 \le n \le 2 \cdot 10^5 $ , $ 0 \le x \le 2 \cdot 10^5 $ , $ 1 \le y \le 2 \cdot 10^5 $ , $ 0 \le s \le 2 \cdot 10^5 $ ) — the length of the sequence, the parameters $ x $ and $ y $ , and the required sum of the sequence elements. The sum of $ n $ over all test cases does not exceed $ 2 \cdot 10^5 $ , and also the sum of $ s $ over all test cases does not exceed $ 2 \cdot 10^5 $ .

For each test case, if the desired sequence exists, output "Yes" on the first line (without quotes). Then, on the second line, output $ n $ integers $ a_1, a_2, \ldots, a_n $ separated by a space — the elements of the sequence $ a $ . If there are multiple suitable sequences, output any of them. If the sequence does not exist, output "No" on a single line. You can output each letter in any case (lowercase or uppercase). For example, the strings "yEs", "yes", "Yes", and "YES" will be accepted as a positive answer.

In the first example, the sequence $ [8, 11, 2, 5, 2] $ satisfies the conditions. Thus, $ a_1 = 8 = x $ , $ a_2 = 11 = a_1 + 3 $ , $ a_3 = 2 = a_2 \bmod 3 $ , $ a_4 = 5 = a_3 + 3 $ , $ a_5 = 2 = a_4 \bmod 3 $ . In the second example, the first element of the sequence should be equal to $ 5 $ , so the sequence $ [2, 2, 2] $ is not suitable.